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LECTURES 


ON 


THE  THEORY  OF  FUNCTIONS  OF 
REAL  VARIABLES 

A^OLUME  I 


BY 

JAMES  PIEEPONT 

Professor  of  Mathematics  in  Yale  University 


'^i^^... 


Math,  dbpt 


GINN  AND  COMPANY 

BOSTON     •    NEW  YORK    •    CHICAGO    •     LONDON 
ATLANTA    •    DALLAS    ■    COLUMBUS    •    SAN  FRANCISCO 


Entered  at  STAxioitEKS'  Hall, 


COPi'KIGHT,  1905 
By  JAMES   PIERPONT 


ALL  RIGHTS   RESERVED 

PRINTED  IN  THE  UNITED  STATES  OP  AMERICA 

529.5 


GINN  A^  D  COMPANY  •  PRO- 
PRIETORS ■  BOSTON  •  U.S.A. 


PREFACE 

The  present  work  is  based  on  lectures  which  the  author  is 
accustomed  to  give  at  Yale  University  on  advanced  calculus  and 
the  theory  of  functions  of  real  variables.  It  falls  in  two  volumes, 
and  the  following  remarks  apply  only  to  the  first. 

The  student  of  mathematics,  on  entering  the  graduate  school  of 
American  universities,  often  has  no  inconsiderable  knowledge  of 
the  methods  and  processes  of  the  calculus.  He  knows  how  to 
differentiate  and  integrate  complicated  expressions,  to  evaluate 
indeterminate  forms,  to  find  maxima  and  minima,  to  differentiate 
a  definite  integral  with  respect  to  a  parameter,  etc.  But  no  em- 
phasis has  been  placed  on  the  conditions  under  which  these  pro- 
cesses are  valid.  Great  is  his  surprise  to  learn  that  they  do  not 
always  lead  to  correct  results.  Numerous  simple  examples,  how- 
ever, readily  convince  him  that  such  is  nevertheless  the  case. 

The  problem  therefore  arises  to  examine  more  carefully  the 
conditions  under  which  the  theorems  and  processes  of  the  calculus 
are  correct,  and  to  extend  as  far  as  possible  or  useful  the  limits  of 
their  applicability. 

In  doing  this  it  soon  becomes  manifest  that  the  style  of  reason- 
ing which  the  student  has  heretofore  employed  must  be  abandoned. 
Examples  of  curves  without  tangents,  of  curves  completely  filling 
areas,  and  other  strange  configurations  so  familiar  to  the  analyst 
of  to-day,  make  it  clear  that  the  rough  and  ready  reasoning  which 
rests  on  geometric  intuition  must  give  way  to  a  finer  and  more 
delicate  analysis.  It  is  necessary  for  him  to  learn  to  think  in  the 
e,  8  forms  of  Cauchy  and  Weierstrass. 

We  have  here  the  beginnings  of  the  theory  of  functions  of  real 
variables,  and  the  twofold  problem  just  sketched  characterizes 
sufticiently  well  the  subject-matter  and  form  of  treatment  of  the 
present  volume. 


iv  PREFACE 

To  obtain  a  foundation,  the  author  has  begun  by  developing  the 
real  number  system  after  the  manner  of  Cantor  and  Dedekind, 
postulating  the  theory  of  positive  integers.  To  obtain  sufficient 
generality,  he  has  employed  from  the  start  the  more  simple  prop- 
erties of  point  aggregates.  No  attempt,  however,  has  been  made 
to  state  every  theorem  with  all  possible  generality.  The  author 
has  allowed  himself  a  wide  liberty  in  this  respect.  Some  theorems 
are  stated  under  very  broad  conditions,  while  others  are  enunciated 
under  extremely  narrow  ones.  Some  of  these  latter  will  be  taken 
up  later  on. 

Two  features  of  this  volume  may  be  mentioned  here.  In  the 
first  place,  the  Euclidean  form  of  exposition  has  been  adopted. 
Each  theorem  with  its  appropriate  conditions  is  stated  and  then 
proved.  Without  doubt  this  makes  the  book  less  attractive  to 
read,  but  on  the  other  hand  it  increases  its  usefulness  as  a  book  of 
reference.  One  is  thus  often  saved  the  labor  of  running  through 
a  complicated  piece  of  reasoning  to  pick  up  sundry  conditions 
which  have  been  introduced,  sometimes  without  any  explicit 
mention,  in  the  course  of  the  demonstration. 

Secondly,  numerous  examples  of  incorrect  forms  of  reasoning 
currently  found  in  standard  works  on  the  calculus  have  been 
scattered  through  the  earlier  part  of  the  volume.  It  is  the 
author's  experience  that  nothing  stimulates  the  student's  critical 
sense  so  powerfully  as  to  ask  him  to  detect  the  flaws  in  a  piece  of 
reasoning  which  at  an  earlier  stage  of  his  training  he  considered 
correct. 

A  few  new  terms  and  symbols  have  been  introduced,  but  only 
after  long  deliberation.  It  is  hoped  that  their  employment  suffi- 
ciently facilitates  the  reasoning,  and  the  enunciation  of  certain 
theorems,  to  justify  their  introduction.  It  may  be  well  to  note 
here  the  author's  use  of  the  word  "  any  "  in  the  sense  of  any  one 
at  pleasure,  and  not  in  the  sense  of  some  one.  The  words  "  each," 
"every,"  "some,"  "any,"  are  often  used  in  an  indiscriminate 
manner,  and  to  this  is  due  a  part  of  the  difficulty  the  beginner 
experiences  in  modern  rigorous  analysis. 

No  attempt  has  been  made  to  attribute  the  various  results  here 
given  to  their  respective  authors.  That  has  been  rendered  un- 
necessary by  the  very  full  bibliographies  of  the  Encyclopddie  der 


PREFACE  V 

Mathematischen  Wissenschaften.  The  author  feels  it  his  pleasant 
duty,  however,  to  acknowledge  his  large  indebtedness  to  the  writ- 
ings of  Jordan,  Stolz,  and  Vallee-Poussin.  He  hopes,  howevei-, 
that  it  will  be  found  that  he  has  not  used  them  servilely,  but  in  an 
individual  and  independent  manner. 

Finally,  he  wishes  to  express  his  hearty  thanks  to  his  friend 
Professor  M.  B.  Porter,  and  to  his  former  pupil  Dr.  E.  L.  Dodd, 
for  the  unflagging  interest  they  have  shown  during  the  composi- 
tion of  this  volume  and  for  their  many  and  valuable  suggestions. 

JAMES   PIERPONT. 

New  Haven,  Conn.,  August,  1905. 


Note 


A  list  of  some  of  the  mathematical    terms   and  symbols  employed  in  this 
woik  will  be  found  at  the  end  of  the  volume. 


CONTENTS 


CHAPTER   I 
RATIONAL  NUMBERS 

ARTICLES  PAGE 

].     Historical  Introduction 1 

2-19.     Fractions 5 

20-30.     Negative  Numbers 12 

31-;>5.     Some  Properties  of  the  System  B 19 

36-39.     Some  Inequalities ^         .         .  22 

40-51.     Rational  Limits 24 


CHAPTER   II 

IRRATIONAL  NUMBERS 

62-53.     Insufficiency  of  JS 31 

54-80.     Cantor's  Theory ' 32 

81-85.     Some  Properties  of  9{ 52 

86-96.     Numerical  Values  and  Inequalities 64 

97-111.     Limits ' 61 

112-122.     The  Measurement  of  Rectilinear  Segments.     Distance      ...  72 

123-127.     Correspondence  between  91  and  the  Points  of  a  Right  Line       .         =  78 

128-131.     Dedekind's  Partitions      ~r 82 

132-143.     Infinite  Limits 85 

144-145.     Different  Systems  for  Expressing  Numbers      .        .        ,         •        .  91 


CHAPTER   III 

EXPONENTIALS  AND  LOGARITHMS 


146-159. 

Rational  Exponents 

.      95 

160-172. 

Irrational  Exponents 

.     101 

173-179. 

Logarithms 

.     109 

180-184. 

Some  Theorems  on  Limits         .... 

.     112 

185. 

Examples 

.     117 

Vlll 


CONTENTS 


CHAPTEE  IV 

THE  ELEMENTARY  FUNCTIONS.     NOTION  OF  A  FUNCTION  IN 

GENERAL 


186-193. 
194-195. 

196. 
197-199. 
200-208. 
209-210. 
211-215. 

216. 
217-220. 
221-224. 


225-229. 
230-237. 
238-239. 
240-241. 


Fdnctions  of  One  Variable 

PAGE 

Definitions * 118 

Integral  Rational  Functions 121 

Rational  Functions 123 

Algebraic  Functions 123 

Circular  Functions 125 

Exponential  Functions 131 

One-valued  Inverse  Functions 131 

Logarithmic  Functions 134 

Many-valued  Inverse  Functions ■  .  135 

Inverse  Circular  Functions 137 

Functions  of  Several  Variables 

Rational  and  Algebraic  Functions 139 

Functions  of  Several  Variables  in  General 143 

Composite  Functions 145 

Limited  Functions 147 


CHAPTER  V 

FIRST  NOTIONS  CONCERNING  POINT  AGGREGATES 

242-253.  Preliminary  Definitions 148 

254-255.  Limiting  Points 157 

256-261.  Limiting  Points  connected  with  Certain  Functions  ....  158 

262-269.  Derivatives  of  Point  Aggregates 162 

270-272.  Various  Classes  of  Point  Aggregates 167 

CHAPTER   VI 


LIMITS  OF  FUNCTIONS 
Functions  of  One  Variable 


273-277.  Definitions  and  Elementary  Theorems 

278-284.  Second  Definition  of  a  Limit    . 

285-294.  Graphical  Representation  of  Limits 

295-305.  Examples  of  Limits  of  Functions 

306-312.  The  Limit  e  and  Related  Limits 


171 

175 
180 
184 
190 


CONTENTS 


IX 


Functions  of  Several  Variables 

ARTICLES  PAGE 

313—317.  Definitions  and  Elementary  Tlieorems 193 

318-321.  A  Method  for  Determining  tlie  Non-existence  of  a  Limit         .        .  196 

322-324.  Iterated  Limits 198 

325-328.  Uniform  Convergence 199 

329-335.  Remarks  on  Diriclilet's  Definition  of  a  Function      ....  202 

336-338.  Upper  and  Lower  Limits 206 


CHAPTER   VII 

CONTINUITY  AND   DISCONTINUITY  OF  FUNCTIONS 

339-342.     Definitions  and  Elementary  Theorems 208 

343-346.     Continuity  of  the  Elementary  Functions 210 

347.  Discontinuity 211 

348.  Finite  Discontinuities 212 

349.  Infinite  Discontinuities 212 

350-358.     Some  Properties  of  Continuous  Functions 214 

369-361.     The  Branches  of  Many-valued  Functions 219 

362.     Notion  of  a  Curve 220 


CHAPTER   VIII 


DIFFERENTIA  TION 


363-364. 
365-366. 
367-371. 
372-389. 
390-392. 
393-404. 
405-408. 
409-413. 


414-417. 
418-422. 
423-430. 
431-433. 
434^435. 


Functions  of  One  Variable 

Definitions 222 

Geometric  Interpretations 223 

Non-existence  of  the  Differential  Coefficient 225 

Fundamental  Formulae  of  Differentiation 229 

Differentials  and  Infinitesimals 244 

The  Law  of  the  Mean 246 

Derivatives  of  Higher  Order 252 

Taylor's  Development  in  Finite  Form 266 

Functions  of  Several  Variables 

Partial  Differentiation 259 

Change  in  the  Order  of  Differentiating 262 

Totally  Differentiable  Functions 268 

Properties  of  Differentials 276 

Taylor's  Development  in  Finite  Form 279 


CONTENTS 


CHAPTER  IX 
IMPLICIT  FUNCTIONS 

AKTIOLBS 

436.     Definitions 

437-438.  Existence  Tiieorems  ;  one  independent  and  one  dependent  variable 

439.     Extension  of  tlie  Domain  of  Existence 

440-443.     Existence  Theorems  ;  several  Variables 


282 
284 
290 
291 


CHAPTER   X 

INDETERMINATE  FORMS 

444-449.  Application  of  Taylor's  Development  in  Finite  Form 

450-451.  The  Form  0/0 

452-153.  The  Form  00 /qo         ..... 

454.  Other  Forms 

455-459.  Criticisms 

460-464.  Scales  of  Infinitesimals  and  Infinities 

465.  Order  of  Infinitesimals  and  Infinities 


298 
301 
305 
307 
308 
312 
316 


CHAPTER   XI 
MAXIMA   AND   MINIMA 

One  Variable 

466-467.     Definitions.     Geometric  Orientation 317 

468-473.     Criteria  for  an  Extreme     .         .         . 318 

474-475.     Criticism 321 

Several  Variables 

476-478.  Definite  and  Indefinite  Forms 322 

479-480.  Semidefinite  Forms .         .         .         .326 

481-483.  Criticism 327 

484-486.  Relative  Extremes 329 


CHAPTER   XII 

INTEGRATION 

487-488.     Geometric  Orientation 333 

489.     Analytical  Definition  of  an  Integral 335 

490-492.     Upper  and  Lower  Integrals       ........  337 

493-498.     Criteria  for  Integrability 340 

499-503.     Classes  of  Limited  Integrable  Functions 344 


CONTENTS 


XI 


ARTICLES  PAGE 

504-508.  Properties  of  Integrable  Functions 346 

509-513.  Functions  with  Limited  Variation 349 

514-518.  Content  of  Point  Aggregates 352 

619-522.  Generalized  Definition  of  an  Integral        , 356 


523-530. 
631-535. 
536-638. 
539. 
640-544. 
545-546. 


647-549. 
550-551. 
652-554. 
565-559. 


560. 
661-566. 

567. 

568. 
569-570. 


CHAPTER   XIII 
PROPER  INTEGRALS 

First  Properties ,        .  361 

First  Tiieorem  of  the  Mean 366 

The  Integral  considered  as  a  Function  of  its  Upper  Limit        .        .  368 
Criticism    ...         ......         .        .         .371 

Change  of  Variable 371 

Second  Theorem  of  the  Mean 377 

Indefinite  Integrals 

Primitive  Functions 380 

Methods  of  Integration 383 

Integration  by  Parts. 384 

Change  of  Variable 386 

Integrals  Depending  on  a  Parameter 

Definitions 387 

Continuity 388 

Differentiation 392 

Integration .  394 

Inversion  of  the  Order  of  Integration 395 


671-577. 
578-590. 
591-605. 
606-607. 
608. 


609-615. 
616-619. 
620-621. 
622-628. 
629-6-31, 


CHAPTER  XIV 
IMPROPER  INTEGRALS.    INTEGRAND  INFINITE 

Preliminary  Definitions     .         .         .* 399 

Criteria  for  Convergence 405 

Properties  of  Improper  Integrals 412 

Change  of  Variable 420 

Second  Theorem  of  the  Mean 421 

Integrals  Depending  on  a  Parameter 

Uniform  Convergence 424 

Continuity 429 

Integration 432 

Inversion 435 

Differentiation 441 


Xll 


CONTENTS 


CHAPTER  XV 
IMPROPER  INTEGRALS.     INTERVAL   OF  INTEGRATION  INFINITE 


ARTICLES 

632-634. 
635-646. 
647-652. 
653-654. 
655-657. 


658-665. 
666-670. 
671-682. 
683-691. 
692, 


PAGE 

Definitions 445 

Tests  of  Convergence 450 

Properties  of  Integrals 456 

Theorems  of  the  Mean .        ,        .  459 

Change  of  Variable 462 

Integrals  Depending  on  a  Parameter 

Uniform  Convergence 464 

Continuity 474 

Integration  and  Inversion 479 

Differentiation 493 

Elementary  Properties  of  B(?<,  u),  r(M)  .        .        .        .        .        .501 


CHAPTER   XVI 
MULTIPLE  PROPER  INTEGRALS 

693.  Notation 506 

694-701.  Upper  and  Lower  Integrals 507 

702-703.  Content  of  Point  Aggregates 512 

704.  Frontier  Points 514 

705-710.  Discrete  Aggregates 515 

711-716.  Properties  of  Content 519 

717-718.  Plane  and  Rectilinear  Sections  of  an  Aggregate       ....  524 

719-721.  Classes  of  Integrable  Functions 526 

722-723.  Generalized  Definition  of  Multiple  Integrals 528 

724-731.  Properties  of  Integrals 531 

732-737.  Reduction  of  Multiple  Integrals  to  Iterated  Integrals       .        .        •  537 

738-740.  Application  to  Inversion  ^ 543 

741-746.  Transformation  of  the  Variable 547 


FUI^OTION  THEORY   OF  REAL 
VARIABLES 


CHAPTER   I 
RATIONAL  NUMBERS 

Historical  Introduction 

1.  The  reader  is  familiar  with  the  classification  of  real  numbers 
into  rational  and  irrational  numbers.  The  rational  numbers  are 
subdivided  into  integers  and  fractions. 

Besides  the  real  numbers  there  is  another  class  of  numbers 
currently  employed  in  modern  analysis,  viz.  complex  or  imaginary 
numbers.  In  this  work  we  shall  deal  almost  exclusively  with 
real  numbers. 

Historically,  the  first  numbers  to  be  considered  were  the  posi- 
tive integers  1,  2,  3,  4,  5,  6,  .  .  .  (3^ 

We  shall  denote  this  system  of  numbers  by  3- 

It  is  not  our  intention  to  develop  the  theory  of  these  numbers ; 
instead,  we  shall  merely  call  attention  to  some  of  their  funda- 
mental properties.* 

In  the  first  place,  we  observe  that  the  elements  of  ^  are  ar- 
ranged in  a  certain  fixed  order ;  that  is,  if  a,  h  are  two  different 
numbers,  then  one  of  them,  say  a,  precedes  the  other  h.  This  we 
express  by  saying  that  a  is  less  than  6,  or  that  h  is  greater  than  a. 

In  symbols 

a<6,         h>a. 

*  For  an  extended  treatment  of  this  subject  we  refer  to  the  excellent  work  of  O  Stolz 
and  J.  A.  Gmeiner,  Theoretische  Arithmetik,  Leipzig,  1900. 

1 


2  RATIONAL  NUMBERS 

We  say  the  system  ^  is  ordered.  Furthermore,  if  a=  b,  5  =  <?, 
then  a  =  e.  Also  ii  a=b  and  b>c,  then  a> c.  Also  if  a > 5, 
b>c,  then  a>c.  Secondly,  we  observe  that  the  system  3'  i^ 
infinite ;  after  each  element  a  follows  another  element,  and  so  on 
without  end. 

On  the  elements  of  3^  we  perform  four  operations,  viz.  addition, 
subtraction,  multiplication,  and  division.  They  are  called  the 
four  rational  operations.  Of  these  operations,  two  may  be  re- 
garded as  direct,  viz.  addition  and  multiplication.  The  other  two 
are  their  inverses,  viz.  subtraction,  the  inverse  of  addition ;  and 
division,  the  inverse  of  multiplication. 

The  formal  laws  governing  addition  are :  the  associative  law, 
expressed  by  the  formula 

a-\-(b  +  e)  =  (a  +  b}-\-c; 

and  the  commutative  law,  expressed  by 

a  +  b  =  b  +  a. 

As  regards  the  position  of  a  +  6  in  the  system  Q,  relative  to  a  or 

6,  we  have  ,  , 

a  +  0  >  a  or  0. 

We  have  also  the  relation 

a-{-b>a'  +  b,  if  a>a'. 

The  formal  laws  governing  multiplication  are  the  three  fol- 
lowing : 

The  associative  law,  expressed  by 

a  •  be  =  ab  •  c. 
The  distributive  law,  expressed  by 

Qa-}-b)c  =  ac  -\-bc,         a(b -\- c")  =  ab  +  ao. 
The  commutative  law,  expressed  by 

ab  =  ba. 
We  have  also  the  relation,  with  respect  to  order, 
ab>a'b  if  a>a'. 


HISTORICAL   INTRODUCTION  3 

« 

Another  important  property  is  this : 

If  ac  =  he,  then  a  =  h. 

The  result  of  subtracting  h  from  a  is  defined  to  be  the  number 
X  in  ^,  satisfying  the  relation 

a  =  h  -\-x. 

But  when  a^h,  no  such  number  exists  in  3^. 

Similarly,  the  result  of  dividing  a  by  6  is  defined  to  be  the 
number  x  in  3,  satisfying  the  relation 

a  =  hx. 

If,  however,  a  is  not  a  multiple  of  6,  no  such  number  exists  in  3. 

Thus  when  we  limit  ourselves  to  the  number  system  ^,  the  two 
operations  of  subtraction  and  division  cannot  always  be  per- 
formed. In  order  that  they  may  be,  we  enlarge  our  number 
system  by  introducing  new  elements,  viz.  fractions  and  negative 
numbers. 

Tlie  introduction  of  fractions  into  arithmetic  was  comparatively 
easy ;  on  the  contrary,  the  negative  numbers  caused  a  great  deal 
of  trouble.  For  a  time  negative  numbers  were  called  absurd  or 
fictitious.  That  the  product  of  two  of  these  fictitious  numbers, 
—  a  and  —5,  could  give  a  real  number,  -{-ah,  was  long  a  stumbling 
block  for  many  good  minds. 

The  introduction  of  irrational  numbers,  i.e.  numbers  like 

V2,     ^5,     7r  =  3.14159...,     e  =  2.7182..., 

never  excited  much  comment.  In  actual  calculations  one  used 
approximate  rational  values,  and  it  was  perfectly  natural  to  sub- 
ject them  to  the  same  laws  as  rational  numbers.  It  is  true  that 
the  Greeks  of  the  time  of  Euclid  were  perfectly  aware  of  the 
difficulties  which  beset  a  rigorous  theory  of  incommensurable 
magnitudes  ;  witness  the  fifth  and  tenth  book  of  Euclid's  Ele- 
ments.  But  these  subtle  speculations  found  little  attention  during 
the  Renaissance  of  mathematics  in  the  seventeenth  and  eighteenth 
centuries.  The  contemporaries  and  successors  of  Newton  and 
Leibnitz  were  too  much  absorbed  in  developing  and  applying  the 
infinitesimal  calculus  to  think*  much  about  its  foundations. 


4  RATIONAL  NUMBERS 

At  the  close  of  the  eighteenth  and  the  beginning  of  the  nine- 
teenth centuries  a  cliange  of  attitude  is  observed.  Gauss,  La- 
grange, Cauchy,  and  Abel  called  for  a  return  to  the  rigor  of  the 
ancient  Greek  geometers.  Certain  paradoxes  and  even  results 
obviously  false  had  been  obtained  by  methods  in  good  repute.  It 
became  evident  that  the  foundations  of  the  calculus  required  a 
critical  revision. 

Abel  in  a  letter  to  Hansteen  in  1826  writes  :  *  "I  mean  to 
devote  all  my  strength  to  spread  light  in  the  immense  obscurity 
which  prevails  to-day  in  anal3^sis.  It  is  so  devoid  of  all  plan  and 
system  that  one  may  well  be  astonished  that  so  many  occupy 
themselves  with  it,  —  what  is  worse,  it  is  absolutely  devoid  of 
rigor.  In  the  higher  analysis  there  exist  very  few  propositions 
which  have  been  demonstrated  with  complete  rigor.  Every- 
where one  observes  the  unfortunate  habit  of  generalizing,  without 
demonstration,  from  special  cases  ;  it  is  indeed  marvelous  that 
such  methods  lead  so  rarely  to  so-called  paradoxes." 

In  another  place  he  writes  :  f  "  I  believe  you  could  show  me 
but  few  theorems  in  infinite  series  to  whose  demonstration  I  could 
not  urge  well-founded  objections.  The  binomial  theorem  itself 
has  never  been  rigorously  demonstrated.  .  .  .  Taylor's  expan- 
sion, the  foundation  of  the  whole  calculus,  has  not  fared 
better." 

The  critical  movement  inaugurated  by  the  above-mentioned 
mathematicians  found  its  greatest  exponent  in  Weierstrass.  It 
is  no  doubt  largely  due  to  his  teachings  that  we  may  boast  to-day 
that  the  great  structure  of  modern  analysis  is  built  on  the  securest 
foundations  known ;  that  its  methods  have  attained,  if  not  sur- 
passed, the  justly  famed  rigor  of  the  ancient  Greek  geometers. 
The  saying  of  D'Alembert,  "  Allez  en  avant,  la  foi  vous  viendra," 
has  lost  its  force.  To-day,  it  is  not  faith  that  is  required,  but  a 
little  patience  and  maturity  of  mind. 

As  Weierstrass  has  shown,  it  is  necessary,  in  order  to  place 
analysis  on  a  satisfactory  basis,  to  go  to  the  very  root  of  the 
matter  and  create  a  theory  of  irrational  numbers  with  the  same 
care  and  rigor  as  contemplated  by  Euclid,  in  his  theory  of  incom- 
mensurable magnitudes,  only  on  a  f?r  grander  scale.  It  is  too 
*  Abel,  (Euvres,  2°  ed.,  Vol.  2,  p.  263.  t  Abel,  I.e.,  p.  257. 


FRACTIONS  5 

early  to  make  the  reader  see  the  necessity  of  this  step,  but  it  will 
appear  over  and  over  again  in  the  course  of  this  work. 

Fractions 

2.  Before  taking  up  the  theory  of  irrational  numbers,  we  wish 
to  develop  in  some  detail  the  modern  theory  of  fractions  and 
negative  numbers.  We  shall  rest  our  treatment  of  these  numbers 
on  the  properties  of  the  positive  integers  3^,  which  we  therefore 
suppose  given.  One  of  these  properties,  on  account  of  its  impor- 
tUnce,  deserves  especial  mention,  viz.: 

If  the  product  ah  is  divisible  by  e,  and  if  a  and  c  are  relatively 
prime,  then  b  is  divisible  by  c. 

3.  Let  us  begin  with  the  positive  fractions.  As  we  saw,  divi- 
sion of  a  by  b,  where  a,  b  are  two  numbers  in  Q,  is  not  possible 
unless  a  is  a  multiple  of  b.  Our  object  is  therefore  to  form  a  new 
system  of  numbers,  call  it  %,  formed  of  the  numbers  of  ^  and  cer- 
tain other  numbers,  in  which  division  shall  be  always  possible. 

We  start  by  forming  all  possible  pairs  of  numbers  in  3-  These 
pairs  we  represent  by  the  notation 

a  =  (a,  a'),    y8  =  (5,  5') — 

In  any  one  of  these  pairs,  as  a  =  (a,  a'),  we  call  a  the  first  con- 
stituent and  a'  the  second  constituent  of  a. 

The  system  g  consists  of  the  totality  of  these  pairs  a,  /3,  ••• 

The  elements  of  g  we  have  represented  by  the  symbol  (a,  a'). 
Any  other  symbol  would  do.  The  customary  ones  are  a/b  and 
a:  b. 

We  have  purposely  avoided  these  symbols,  so  familiar  to  the 
reader,  in  order  that  his  attention  shall  be  more  closely  fixed  on 
the  logical  processes  employed. 

4.  The  objects  of  5  have  as  yet  no  properties  ;  we  proceed  to 
assign  them  one  arithmetic  property  after  another,  taking  care 
that  no  property  shall  contradict  preceding  ones.  We  begin  by 
setting  5  in  relation  to  ^.  We  say  :  (a,  a')  shall  be  a  number  c, 
in  3^?  when  a  =  a'c.  Thus,  any  element  of  ^  whose  first  constituent 
is  a  multiple  of  its  second,  is  an  element  of  ^,  i.e.  a  positive  integer. 


6  RATIONAL   NUMBERS 

From  this  follows  that  every  number  a  of  -3^  lies  in  ^.  For, 
(a,  1)  lies  in  g.     On  the  other  hand 

(a,  1)  =  a. 
Hence  a  lies  in  ^. 

5.  We  define  next  the  terms,  equals  greater  than,  less  than. 
Let  «=(«,  a'),  /3=(^',  b'). 

We  say  :  a  =  /3  according  as  ah'  =  a'h. 

We  observe  that  to  decide  the  equality  or  inequality  of  two 
elements  in  5,  the  operations  required  are  on  the  elements  of  Q. 

6.  We  deduce  now  some  of  the  consequences  of  the  above  defi- 
nition of  equality  and  inequality.  In  the  first  place,  suppose  a,  /3 
both  lie  in  3^ ;  i.e.  let 

a  =  (aa',  a')  =  a,  y8  =  (hh\  J')  =  J, 
by  4.     Now,  according  to  the  definition  in  5, 


according  as 

that  is,  according  as 


«|^ 


aa'b'  =  a'bb'; 


a^b. 


Thus,  when  a  considered  as  a  number  of  -3^,  equals  /3  considered 
as  a  number  of  Q,  the  two  are  equal,  considered  as  numbers  in  ^, 
and  conversely. 

7.    If  «  =  7?  y8  =  7i  the7i  a  =  0. 

For,  let  a  =  (a,  a'),  /3  =  (J,  b'),  7  =  (c,  c'). 

Since  «  =  7,  ac'  =  a'c,  by  5.  (1 

Since  /8=7,  bc'  =  b'c.  (2 

Multiply  1)  by  6',  and  2)  by  a'  and  subtract. 

Then  ah'c'  =  a'Jc'. 

.-.  ah'  =  a'h.      .-.  a=  8, 
by  5. 


FRACTIONS  7 

8.  The  two  numbers  {ma,  ma'^  and  (a,  a'')  are  equal. 

This  follows  at  once  from  the  definition  in  5.  From  this  fact 
we  conclude :  We  can  multiply  the  first  and  second  constituent  of  a 
number  without  changing  its  value. 

Conversely : 

If  the  first  and  second  constituents  of  a  number  have  a  common 
factor,  it  can  be  removed  without  changing  the  value  of  the  7iumber, 

9.  Let  a,  a'  be  relative  prime.  For  a  =  (a,  a')  and  /3  =  (5,  6') 
to  be  equal,  it  is  necessary  and  sufficient  that 

b  =  ta,  b'  =  ta'.  (1 

Obviously  if  1)  holds,  a  =  j3.     The  condition  1)  is  thus  sufficient. 
It  is  necessary.     For,  from  «  =  /S,  we  have 

ab'  =  a'b.  (2 

We  apply  now  the  property  mentioned  in  2.  Since  a'b  is 
divisible  by  a,  by  virtue  of  2) ;  and  since  a,  a'  are  relative  prime, 
b  must  be  divisible  by  a.     Say 

b  =  ta.  (3 

Similarly,  since  ab'  is  divisible  by  a',  and  a,  a'  are  relative  prime, 
b'  must  be  divisible  by  a'.     Say 

b'  =  sa'.  (4 

Putting  3),  4)  in  2),  we  get  s  =  t. 
Hence  3),  4)  give  now  1). 

10.  Our  next  step  is  to  define  the  four  rational  operations  on 
the  elements  of  g. 

We  begin  by  defining  the  two  direct  operations. 

Let  «  =  (a,  a'),  /3  =  (5,  J'). 

We  define  addition  by  the  equation, 

a  +  l3=(ab'  +  a'b,  a'b'');  (1 

and  multiplication,  by 

a^  =  (^ab,  a'b'y.  (2 


8  RATIONAL   NUMBERS 

It  can  be  shown  that  the  operations  just  defined  enjoy  the  same 
properties  as  those  of  ordinary  fractions.  Without  stopping  to 
show  this  in  detail,  we  demonstrate  a  few  of  these  properties,  by 
way  of  illustration. 

11.  Let  a,  ^  lie  in  ^  ;  and  say 

a  =  {aa\  a')  =  a,  /3  =  {bb\  5')  =  h. 

Then  a  +  y8,  as  defined  in  10,  1),  should  give  a  +  6  ;  and  ayS,  as 
defined  in  10,  2),  should  give  ab. 
This  is  indeed  so.     For 

a  +  ^  =  (iaa'b'  +  a'bb',  a'b'),  by  10,  1) 

=  («  +  J,  1),  by8 

=  a  +  5,  by  4. 

Similarly, 

a-l3  =  (aa'bb',  a'b'),  by  10,  2) 

=  (a5,  1),  by8 

=  ab,  by  4. 

12.  From  ay  =  fiy,  follows  a  =  /S. 
For,  let  7  =  (e,  c') ;  we  have  : 

«7  =  (a,  a')(c,  e')  =  (ac,  «'c'),  by  10,  2) 
^y=(b,b'Xc,c'}  =  (bc,b'c'}. 
Since  by  hypothesis  ay  =  ^y,  we  have 

acb'c'  =  a'c'bc,  by  5. 

.-.  ab'  =  a'b. 

.'.  «=  A  by  5. 

13.  We  establish  now  the  following  relations: 

1)  a  +  /3>«. 

2)  If  yS>7,  then  a-{-/3>a  +  y. 

3)  If  a  +  /3  =  a  +  7,  then  ^  =  y. 

4)  If  a>y3  and  /3>7,  then  «>7. 


FRACTIONS  9 

To  prove  1): 

«  +  ;8  =  (a,  a')  +  Q>.  ^')  =  (ab'  +  a'b,  a'b'),  by  10,  1). 
But  a'(ab'  +  a'b)>aa'b'. 

.'.  a  +  /3  >  «,  by  5. 

To  prove  2): 

a  +  ^  =  (ab'  +  a'b,  a'V) 

=  (ab'c'  +  a'bc\  a'b'c'),  by  8.  (4 

Similarly, 

«  +  ry  =  (ac' +  a'c,  a'c') 

=  {ab'c' +  a'b'c,  a'b'c'y  (5 

By  hypothesis  yS  >  7 ;  hence,  by  5, 

Jc'  >  b'c.  (6 

Comparing  4),  5),  we  see  the  second  constituents  are  equal, 
while  the  first  constituent  in  4)  is  greater  than  the  first  constituent 
in  5),  by  virtue  of  6).  From  this  follows,  by  5,  that  a  +  yS>a  +  7, 
which  is  2). 

To  prove  3): 

Suppose  the  contrary  ;  then  since  /3  9^  7,  either  /3  >  7  or  y8  <  7. 
If  /3>7,  then  «  +  /8>«  +  7,  by  2).  (7 

If  /3<7,  then  a  +  7>«  +  yg,  by  2).  (8 

But  both  7),  8)  contradict  the  hypothesis  that  a  +  yS  =  a  +  7. 


(9 
(10 


To  prove  4) : 

Since 

«>/3, 

ab' 

>a'b. 

Since 

/3>7, 

be' 

'>b'c. 

From 

9),  10) 

we 

have 

abb'c' 

>a'bb'c; 

whence 

ac' 

>a'c. 

Hence 

',  by  5, 

a 

:>y. 

10  RATIONAL   NUMBERS 

14.  As  an  illustration  of  the  demonstration  for  the  formal  laws 
governing  addition  and  multiplication,  let  us  show  that  the  dis- 
tributive law  holds  in  g.      We  wish  to  prove  that 

1)  a(/S  +  7)  =  «/3  +  ay. 

Now  /S  +  7  =  (be'  +  b'c,  b'c%  by  10,  1). 

a{l3  +  7)  =  (a,  a')  •  (be'  +  b'c,  h'c') 

2)  =  {abc'  +  aJ'c,  a'b'c%  by  10,  2). 

Also 

a^  =  (ab^  a'b');  ay  =  Qac,  a'c'^. 

.'.  a^ -\- ay  =  {aa'bc' -\- aa'b'e,  a'Wc'') 

3)  =  (abc'  +  ab'c,  a'b'c'},  by  8. 
The  comparison  of  2),  3)  gives  1). 

15.  We  turn  now  to  the  inverse  operations,  subtraction  and 
division  ;  considering  first  division. 

We  define  the  quotient  of  a  by  /8  to  be  the  element  or  elements, 
I,  if  any  exist,  of  ^  which  satisfy  the  relation 

«  =  /3|.  (1 

Set  f  =  Qx,  a;').     Since  |  must  satisfy  1),  we  iiave 

(a,  a')  =  (^  b')(ix,  x')=::(bx,  b'x'). 

The  first  and  third  members  give,  by  5, 

ab'x'  =  a'bx,  (2 

which  a:,  x'  must  satisfy. 

A  solution  of  2)  is  obviously 

x=ab',         x' =  a'b. 
Thus, 

is  a  solution  of  1).     This  is  the  only  solution  of  1).     For,  sup- 
pose ?;  is  a  solution.     Then  by  definition 

a  =  ^7).  (3 

Then  1),  3)  give,  by  7, 

which  gives,  by  12, 


FRACTIONS  11 

16.  The  quotient  of  «  by  /3,  we  shall  now  represent  b)''  a//3. 
All  the  numbers  of  ^  may  be  regarded  as  quotients  of  numbers 
in  Q.  For,  let  a=(a,  a')  be  any  number  of  g.  It  evidently 
satisfies  the  equation  ^        t^ 

which,  as  we  have  just  seen,  admits  only  one  root,  viz.  the  quotient 
of  a  by  a'. 

Hence  a  =  (a,  a')  =  a/a'. 

Thus  the  elements  or  numbers  in  ^  are  ordinary  positive  frac- 
tions. 

17.  We  have  now  this  result.  In  the  system  ^,  division  is 
always  possible  and  unique.  In  the  old  system  Q,  this  is  not 
true ;  the  division  of  a  by  J  being  only  possible  when  a  is  a 
multiple  of  b.  We  see,  then,  that,  on  properly  enlarging  our 
number  system  by  introducing  new  elements,  we  obtain  a  system 
1^  which  has  this  advantage  over  3,  that  the  quotient  of  any  two 
numbers  in  ^  exists  and  is  unique. 

18.  We  treat  now  subtraction. 

We  define  the  result  of  subtracting  /3  from  a  to  be  the  element 
or  elements,  call  them  |,  in  ^,  which  satisfy  the  relation 

«  =  yS  +  ^  (1 

7f /3  ^  a,  there  exists  no  number  ^  in  ^  which  satisfies  1).      For, 

'^"  ^  "  '''  /S  +  I  =  «  +  I  >  «,  by  13,  1). 

If  /3  >  «,  /3  +  I  >  «  +  I,  by  13,  2). 

Also  a-\-  ^>  a,  by  13,  1). 

.-.  /S  +  I  >  «,  by  13,  4). 

Thus  when  /3  ^  «,  /S  -f-  f  >  «,  and  hence  /3  +  |  =^  a. 
Suppose  then,  that  /3  <  «.     Then 

ab'  >  a'b.  (2 

From  1),  we  have,  setting  ^  =  (x,  x')  ; 

ia,a')  =  (6,  b')  +  (ix,x') 

=  Cbx'  +  b'x,b'x'),hylO,f), 


12  RATIONAL   NUMBERS 

Hence  by  5,  observing  2), 

a'b'x  =  x'(ab'  -a'b^.  (3 

A  solution  of  3)  is  evidently 

X  =  ah'  —  a'h,  x'  =  a'h'. 
Hence 

I  =  (a5'  _  a'h,  a'h') 
is  a  solution  of  1). 

This  is  the  only  solution;  for  if  ?;  is  a  solution,  w§  have,  by 

definition,  ^  ,  ,, 

a  =  y8  +  7?.  (4 

The  comparison  of  1),  4)  gives 

yS  +  I  =  /S  +  .;. 
Hence,  by  13,  3), 

19.  We  have  thus  this  result :  In  the  system  5,  the  subtraction 
of  yS  from  a  is  possible  and  unique,  when  a  >  /8 ;  when  a  ^  /3,  it  is 
impossible.  That  is,  there  is  no  number  |  in  %  which  satisfies  18, 
1).  When  subtraction  is  possible,  we  represent  the  result  of  sub- 
tracting y8  from  a  by  a  —  yS. 

Negative  Numbers 

20.  In  the  system  of  positive  fractions  %,  subtraction  is  only 
possible  when  the  minuend  is  greater  than  the  subtrahend.  To 
remove  this  restriction,  we  propose  to  form  a  new  number  system 
_B,  which  contains  all  the  numbers  of  ^  ;  and  in  which  subtraction 
of  a  greater  from  a  less  shall  be  possible.  Since  the  method  of 
forming  M  is  identical  with  that  employed  for  g,  we  shall  be  more 
brief  now.  The  numbers  in  ^  we  now  denote  hy  a,  b,  c,  .  .  .  , 
while  the  Greek  letters  a,  /3,  y,  .  .  .  shall  denote  numbers  in  the 
new  system  M. 

21.  1.  We  begin  by  taking  the  elements  of  ^  in  pairs,  to  form 
new  objects,  which  we  de'note  by  the  new  symbol  {a,  b\.  The 
totality  of  all  such  pairs  forms  the  system  R. 

Next,  we  place  H  in  relation  to  g.  Let  a  =  \a,  bl.  In  case 
a  >  b,  we  say  a  shall  be  the  number  a  —  b,  which  obviously  lies  in 


NEGATIVE   NUMBERS  13 

5.     Thus  every  number  in  g  lies  in  R.     For,  let  a  be  any  number 
in  g,  and  let  h  be  any  other  number  in  g.     Then 

\a  -\-  h^h\  =  (^a  +  b')  —  b  =  a; 
that  is,  a  lies  in  jR. 

2.  We  next  order  the  system  i2.     We  say 

\a,a'i>\b,b'l, 
according  as 

a  +  b'fa'  +  b.  (1 

3.  Addition  is  defined  by  the  relation 

a  +  0=  \a,  a' I  +  {b,  h'\  =  ja  +  J,  «'  +  b'\.  (2 

Multiplication  is  defined  by 

«./3  =  ^a5 +a'J',  «J' +  a'i|.  (3 

4.  As  a  consequence  of  1),  we  have 

la,  a'l  =  {a  +  b,a'  +  b\,  (4 

where  b  is  any  number  in  ^. 

In  words,  4)  states  : 

We  can  add  the  same  number  b  to  both  constituents  of  a  =  \a,a'\ 
without  changing  the  value  of  a;  and  if  a,  a'  are  both  >  b,  we  can 
subtract  b  from  both  constituents,  without  altering  the  value  of  a. 

5.  It  is  easy  now  to  prove  results  analogous  to  those  in  6,  7, 11, 
14  ;  in  particular  the  associative,  commutative,  and  distributive 
laws. 

22.  According  to  our  definition  of  equality,  all  the  elements 
of  R  whose  first  and  second  constituents  are  the  same,  i.e.  all 
elements  of  the  type 

\a,  a\, 
are  equal.     We  set 

fa,  a|=0, 
and  call  this  number  zero. 
Then,  if  in  a  =  |  a,  a'  | , 

a>  a',  a  >  0  ; 

if  a'  >  a,  a  <  0. 


14  RATIONAL  NUMBERS 

Numbers  in  R  which  are  >  0,  are  called  positive  ;  those  <  0,  are 
called  negative.  The  number  0  is  neither  positive  nor  negative. 
From  this,  it  follows  that  the  positive  numbers  of  R  are  simply 
the  numbers  of  g;  while  0  and  all  negative  numbers  do  not  lie 
in  g. 

23.  1.  We  observe  that 

«+0  =  «=0  +  «,  (1 

a  .  0  =  0  =  0  •  a.  (2 

To  prove  1). 

Let  a=  \a,  a'l,  0=  lb,bl. 

Then  a+0  ^  \a +  b,  a' +  b\,  by  21,  d; 

=  la,  a'|,by21,  4; 

=  a. 

To  prove  2). 

a  •  0  =  \ab  -\-  a'b,  ab  +  a'b\ 

=  0,  by  22. 

2.    We  also  note  the  relations 

0  +  0  =  0,    0-0  =  0. 

24.  We  can  prove  now  easily  the  Rule  of  signs.  TJie  product  of 
tzvo  positive  or  two  negative  numbers  is  positive.  The  product  of  a 
positive  and  a  negative  number  is  negative. 

Let  «=  \a,  a'\,  ^=  \b,  b'\. 

1°.  «,  /8  >  0,   then  a^  >  0. 

For  here  a> a',  b>b',  hy  22. 

.:  a/3=  \ab  +  a'b\  ob'  +  a'b\  =  {b(a  ~  a')-\- a'b\  ab'\,  by  21,  4 

=  lb(a-  a'),  6'(a  -  a')  I,  by  21,  4 

>0,  since  b{a  ^  a'^>b'(a  —  a'). 

2°.  «,  /3<0,  then  «/3>0. 

Here  a'>a,  b' >b,  by  22. 


NEGATIVE   NUMBERS  15 

/.  a^=\ah^a'(h'  -h-),  ab' U  by  21,  4; 
=  la'(6'-6),  aCb'-b^l,  by  21,  4; 
>0,  since  a'(b'  -b)>a(b'  -h^. 

3°.  a>0,  y8<0,  then  «/3<0. 

Here  a>a\  b'>b. 

.-.  a/8=fa6  +  a'(5'-5),  ab'\ 

=  \a'(b'-b~),  a(b'-by\    ■ 

<0,  since  a' (b'  -b)<a(b'  -b). 

25.  The  product  of  any  two  numbers  in  R  vanishes  when,  and 
only  when,  one  of  the  factors  is  zero. 

Let  a,  /3  be  any  two  numbers  in  R. 

We  saw  in  23  that 

«/3=0, 
when  either  a  or  /3  =  0. 

Conversely,  if  ayS=  0,  either  a  or  /3  =  0. 

For,  if  neither  a  nor  /3  =  0,  these  numbers  are  either  positive  or 
negative.  Their  product  is  therefore  either  positive  or  negative 
by  24,  and  hence  not  zero.     This  is  a  contradiction. 

26.  Let  us  consider  the  following  important  formulae,  viz. : 

1)  If  /3  >  7,  then  a  +  /8  >  «  +  7. 

2)  From  a  +  /3  =  «  +  7,  follows  /S  =  7. 
8)  If  a  =5^  0  and  a/3  =  «7,  then  yS  =  7. 

To  prove  1): 

a-f./3=  \a  +  b,  a'  +  b'\,  by  21,  3  ; 

=  {«  +  ^  +  c',  a'  +  5'  +  c'|,  by  21,  4.  (4 

Similarly, 

a  +  7=  fa  +  c,  a'  +  c'(  =  fa  +  6'+c,  «'  +  5'  +  c?'|.  (5 

Since /3>  7,  J  +  ,' >  6' +  .,  by  21,  2. 

.-.  a  +  J  +  c'  >  a  +  6'  +  c.  (6 


16  RATIONAL   NUMBERS 

If  we  now  apply  the  definition  for  greater  than  given  in  21,  2 
to  «  +  /3  and  a  4-  7,  the  relations  4),  5),  6)  show  that  «  +  yS  >  a  +  7. 

To  prove  2): 

Suppose  the  contrary,  i.e.  suppose  /3  >  7  or  /3  <  7. 

If  /3>7,  «  +  ;S>«  +  7,  by  1). 

If7>/3,  «  +  7>«  +  /3,  by  1). 

Thus  in  both  cases,  a  +  /3  ^t  a  +  7,  which  is  contrary  to  hypothe- 
sis.    Hence  /S  =  7. 

To  prove  3) : 

From  a/S  =  057  we  have 

a(/3-7)=0. 

Applying  25,  we  have  /3  =  7. 

27.  We  turn  now  to  subtraction.  This  we  define  as  in  5,  viz. : 
the  result  of  ^btracting  /3  from  «  is  the  element  or  elements  |,  of 
R.  which  satisfy  0  ,   t-  /-1 

This  equation  gives,  setting  |=  fa;,  x'|, 

{a,  a'|=  16,  6'f  +  fa;,  a;'^  =  f5  +  a:,  h'  +  x'\^hy21,  3. 

Hence  by  21,  2, 

a  +  6'  +  a;'=a'  +5  +a;. 

This  equation  is  evidently  satisfied  by 

x  =  a  +  h\  x'  =  a'  +  h. 

Hence  <.,■,,.-,. 

^=\a  +  h\  a'  +  h\  (2 

is  a  solution  of  1). 

This  is  the  only  solution.     For,  let  ?/  be  a  solution.     Then  by 

definition,  a  ,  ^o 

The  comparison  of  1},  3)  gives 
Hence  by  26,  2),  ^ 


NEGATIVE  NUMBERS  17 

28.  We  have  thus  this  result :  in  the  system  R  subtraction  is 
always  possible  and  unique.  The  result  of  subtracting  /3  from  a, 
we  represent  by  a  —  /3;  it  is  a  number  in  R.  Then  any  number 
«=  {a,  a'\  in  R,  is  the  result  of  subtracting  a'  from  a,  or 


For, 

Similarly, 

Hence 


a=  a  —  a. 
a  =  \a  +  h,  h\,  by  21,  4. 

a'  =  \a'  +  h,  h\. 
a  —  a'  =  \a  +  h,h\  —  \a'  -\-h,h\ 

=  {a  +  25,  a'  +  26|,by27,  2) 
=  fa,  a'|,by21,  4. 

29.    1.    Let  a  =  fa,  a'\  be  any  number  of  R. 

The  number  fa',  a\  is  called  minus  «,  and  we  write 

fa',  a|  =  —  «. 

—  (—  «)=  —  fa',  a|  =  fa,  a'j  =  a. 

^^^'''  «  +  (-«)=  fa,  a'|  + fa',  a| 

=  f  a  +  a',  a  +  «'  I  =  0 

=  «—  a. 

If  a  is  positive,  —  a  is  negative  ;  and  conversely,  if  a  is  nega- 
tive, —  a  is  positive. 

2.  The  number  —a  may  be  defined  as  the  number  |,  such  that 

«  +  |  =  0. 

For,  f  =  —  a  satisfies  this  equation  ;  and,  as  we  saw  in  27,  this 
equation  admits  but  one  solution.  This  shows  that  the  numbers 
in  i2,  =f=  0,  may  be  grouped  in  pairs,  such  that  their  sum  is  zero. 

3.  If   -  a  =  -  /3,  then  a  =  /3. 

For,  multiplpng  both  sides  of  —  a  =  —  yS  by  —  1,  we  get  a  =  yS. 

4.  Every  number  a,  of  i?,  different  from  zero,  can  be  written 

in  the  form 

«  =  a,  or  a—  —  a^) 

where  a  is  a  number  in  5. 


18  RATIONAL   NUMBERS 

For  if  a  >  0,  we  already  know  by  22  that  a  is  a  number  in  ^. 
If  a  <  0,  then  —  «  is  positive,  so  that  —  «  =  a,  a  number  in  ^. 
Multiplying  this  equation  by  —  1,  we  get 

oc  =  —  a. 

30.     1.    We  treat  now  division. 

We  say :  the  result  of  dividing  «  by  /3  is  the  number  or  num- 
bers P,  of  M,  such  that 

«=I/S.  (1 

Suppose  /3^0;  then  there  is  one  and  only  one  number  f ;  i.e.  in 
this  case^  division  is  possible  and  unique. 

There  can  be  at  most  one.    For,  if  t]  satisfies  1),  we  should  have 

a  =  v/3.  (2 

Comparing  1),  2),  we  have 

1/3  =  7;^; 
whence  by  26,  3), 

To  show  that  there  is  always  one  solution  of  1),  we  have  the 
following  cases. 

Let  a,  /3  >  0 ;  then  a  =  a,  /3  =  6,  by  29,  4 ;  and  1)  becomes 

a  =  |5.  (3 

But  by  15  the  solution  of  3)  is 

I  =  (a,  6)  =  a/b. 
Let  a,  /S  <  0  ;  then 

a  =  -  a,     y8  =  -  6,  by  29,  4. 
Then  1)  becomes 

-  a  =  -b^', 
or  by  29,  3, 

Hence  as  before, 

Let  a  >  0,  yS  <  0 ;  then  a  =a,  ^  =  —  b,  and  1)  becomes 

a  =  -bl  (4 


SOME   PROPERTIES   OF   THE   SYSTEM  K  lU 


Set  -1=7?; 

then 

4)g: 

ives 
a  = 

:  hrj. 

Hence, 
id 

V  = 
1  = 

a/b 
■  -  a/b. 

lfa<0,and^>0 

,  we 

get  again 

^  = 

■■  -  a/b. 

Finally,  let  a 

=  0; 

then 

1)  becomes 

Hence  by  25, 

0  = 

:0. 

2.    We  consider  now  the  case  that  /3  =  0. 

The  equation  1)  admits  now  no  solution,  unless  a  =  0  also. 
For,  when  yS  =  0,  /3|  =  0,  whatever  f  may  be. 

If  now  a  =  0,  the  equation  1)  is  satisfied  for  every  number  f  in 
H.  We  have  thus  this  result :  When  the  divisor  is  zero,  division 
is  either  impossible  or  entirely  indeterminate. 

For  this  reason,  division  by  zero  is  excluded  in  modern  mathe- 
matics. The  admission  of  division  by  zero  by  the  older  mathe- 
maticians, Euler  for  example,  has  caused  untold  confusion.  We 
shall  see  it  is  entirely  superfluous. 

Some  Properties  of  the  System  R 

31.  The  system  B,,  which  we  have  just  formed,  is  made  up  of 
the  totality  of  positive  and  negative  integers  and  fractions,  and 
also  zero.  It  is  called  the  system  of  rational  numbers  ;  any  element 
in  it  being  called  a  rational  number.  The  elementary  arithmetical 
properties  of  these  numbers  having  been  established,  there  is  no 
further  occasion  to  employ  the  special  notations  {a,  6)  and  \a,  b\\ 
we  shall,  instead,  employ  the  customary  ones.  Furthermore,  we 
shall  represent  for  the  rest  of  this  chapter  the  numbers  in  B, 
indifferently  by  Greek  and  Latin  letters  a,  b,  c,  •••  a,  y8,  7,  ••• 

For  the  sake  of  completeness  we  now  proceed  to  deduce  a  few 
properties  of  R,  although  the  reader  is  probably  familiar  with  them. 


20  >  RATIONAL  NUMBERS 

32.  The  system  R  is  invariant  with  respect  to  the  four  rational 
operations. 

This  simply  means  that  the  addition,  subtraction,  multiplica- 
tion, and  division  of  any  two  elements  of  i2,  division  by  0  of 
course  excluded,  always  leads  to  an  element  in  R. 

We  saw  this  is  not  true  for  the  systems  ^  and  %. 

33.  1.    The  system  R  is  dense. 

This  term,  taken  from  the  theory  of  aggregates,  which  we  shall 
take  up  later,  simply  means  that  between  any  two  numbers  a,  h 
in  R.,  exists  a  third  and  hence  an  infinity  of  numbers. 

For,  let 

and  say  a>h. 

Then 

d  =  a^Jg  ~  ^2^1 
is  an  integer  >  1. 

Let  e  be  a  positive  integer.     By  taking  it  large  enough,  we  can 

make 

ed>n, 

where  n  is  an  arbitrarily  large  positive  integer. 
Let  h  be  any  integer,  such  that 

ea^-^  <h<  eajb^. 

Then 

h 

lies  between  a,  b  and  represents  at  least  n  numbers. 

2.  The  system  Q  is  not  dense.  For,  if  we  take  a  =  n  and  5  =  w  +  l, 
no  element  of  Q  lies  between  a  and  b. 

From  this  results  a  remarkable  difference  between  Q  and  R. 
After  any  element  n  ol  Q  follows  a  certain  next  element,  viz.  n  +  1. 
Not  so  in  R.  If  a  is  any  number  of  R,  there  exists  no  next  num- 
ber to  a.  For,  if  b  were  that  number,  there  would  lie  no  number 
of  R  between  a  and  b.  But  since  R  is  dense,  there  lie  an  infinity 
of  numbers  of  R  between  a,  b. 


SOME  PROPERTIES  OP  THE   SYSTEM  R  21 

34.  1.  The  system  R  is  an  Archimedian  system.  That  is,  there 
is  no  positive  number  a  in  R  so  small  but  that  some  multiple  of 
a,  say  na,  is  greater  than  any  prescribed  positive  number  h  of  R. 

For,  let 


a-, 
a„ 


Let  us  choose  n  so  large  that 
Then 


n  >  a^b. 


na-i      a-,aj)  ^      7-^7 
na  =  — 1  >  -1-2-  >  a^  >  0. 

2.    Let  a  he  an  arbitrarily  large  number  of  R;  there  exists  a  posi- 
tive integer  n,  such  that  a/71  <  b,  where  b  is  arbitrarily  small. 
For,  by  1,  there  exists  a  positive  n,  such  that 

nb>a. 
Hence 

a/n<b. 

35.    Let  us  lay  off  the  numbers  of  .R  on  a  right  line  L,  just  as  is 
done  in  analytic  geometry. 

: 1 1 1 iS 


Thus,  having  chosen  an  arbitrary  point  0  as  origin,  and  an  arbi- 
trary segment  OU  as  the  unit  of  length,  to  the  positive  number 
p  in  i2,  corresponds  the  point  P  on  i  to  the  right  of  0,  and  at  a 
distance  p  from  0.  To  the  negative  number  —p,  corresponds  the 
point  Q,  lying  to  the  left  of  0,  and  such  that  OQ=p.  To  zero 
corresponds  the  origin  0. 

Let  a,  6,  e,  •••  be  numbers  of  R,  to  which  correspond  the  points 
A,  B,  C,  •••  on  L.  If  a<b,  the  point  B  lies  to  the  right  of  A; 
a  a<b<c,  the  point  B  lies  between  A  and  O. 

The  point  A,  corresponding  to  the  number  a,  is  called  the 
representation  or  image  of  a. 

The  points  corresponding  to  the  numbers  in  R  we  call  rational 
points. 

Then,  since  R  is  dense,  the  aggregate  %  formed  of  the  totality 
of  rational  points  is  also  dense.     That  is,  if  P,  Q  be  any  two  points 


22  RATIONAL   NUMBERS 

of  21,  there  lie  an  infinity  of  points  between  P,  Q  which  belong 
to  21. 

Furthermore,  if  P  be  any  point  of  21,  there  is  no  next  point  to  P. 

This  representation  of  the  numbers  of  R  by  points  on  a  right 
line  is  of  great  assistance  to  us  in  our  reasoning,  as  we  shall  see. 

SoTne  Inequalities 

36.  We  have  seen  in  29,  4  that  every  number  a^O  in  i2,  may 

be  written 

a=  ±05^, 

where  a^  is  a  positive  number. 

The  numerical  or  absolute  value  of  a  is  a^,  and  is  denoted  by 

\a\. 
We  have  thus, 

\a\  =  a^. 
We  also  set 

|0|  =  0. 

For  example  : 

I 41—4.  I_l4|  —  4. 

|3-7|  =  |7-3|  =  4. 

37.  1.  We  have  now  the  following  fundamental  relations : 

|a|  =  |-a|;  (1 

I  a  —  5 1  =  1 5  —  a  I ;  (2 

\a±b\<\a\-\-\b\;  (3 

I  a  ±  J I  ^  1 1  a  I  —  I J 1 1 ;  (4 

I a6 1  =  I  a  I • 1 6 1 ;  (5 


^1         J^O.  (6 

b\ 


They  are  readily  proved.     For  example,  consider  3). 
There  are  various  cases,  according  as  a,  b  are  positive,  negative, 
or  zero.     We  treat  one  specimen  case. 


SOME  INEQUALITIES  23 

Let        a>0,  h<0.  Let         h  =  -bQ,  5o>0. 

Then  a  +  h  =  a  —  b^,     a  —  b  =  a  +  Bq. 

li  a>bQ,  \a-{-b\^\a  —  bQ\=a  —  bQ<a  +  bQ  =  \a\  +  \b\. 

Ifa<5(j,   \a  +  b\  =  \a  —  bQ\=  b^  —  a<a-i-bQ  =  \a\  +  \b\. 

\a  —  b\  =  \a  +  bQ\  =  a  +  bQ  =  \a\  +  \b\. 

Which  establishes  3)  for  this  case.     The  other  cases  are  treated 
similarly. 

2.  By  repeated  applications  of  3),  5)  we  get 

\a^±a^±---±a„,\<\a^\-\ l-|a^|  ;  O 

|ai  •a2  ••' «m|  =  |«il  |«2l  •••  l«m|-  (8 

3.  Let  J.>0,     and     \a\<A.  (9 

Then  from  9)  follows 

-A<a<A;  (10 

and  conversely  from  10)  follows  9). 

38.  1.   An  important  relation  is  the  following : 

Let  \a-b\<A,      \b-c\<B.  (1 

^^^**  \a-c\<A  +  B.  ■  (2 

For,  from  1)  we  have,  by  37,  3, 

—  A<a  —  b<A; 

-B<b-c<B. 

Adding, 

-{A  +  B^<a-c<A  +  B, 
which  gives  2). 

2.   A  special  but  common  case  of  the  above  is  when  A  =  B\  then 

\a-c\<2A.  (3 

We  shall  say  that  2)  or  3)  is  obtained  from  V)  by  adding. 

39.  It  will  be  useful  to  bear  in  mind  the  geometric  interpreta- 
tion or  image  of  certain  inequalities  which  recur  constantly  in  the 
following. 


24  RATIONAL  NUMBERS 

Let  a  be  an  arbitrary  rational  number. 


a-b   a      a+h 
H 1 1 


On  either  side  of  the  point  a  let  us  mark  off  points  a  —  h,  a  +  b, 
at  a  distance  b  from  a.  The  rational  numbers  x  whose  images 
fall  in  the  interval  a—  b,  a  +  b,  evidently  satisfy  the  relation 

a  —  b<x<a  -\-b; 
or  what  is  the  same, 

—  b<x  —  a<b. 

That  is,  X  satisfies  the  inequality 

\x-a\<b.  (1 

Conversely,  the  images  of  the  rational  numbers  x  which  satisfy 
1)  lie  in  the  interval  a  —  b^  a  +  b. 

Similarly,  the  images  of  the  rational  numbers  x  which  satisfy 

0<x-a<b, 

lie  in  the  interval  a,  a  +  b;  while  those  corresponding  to 

0<a-x<b 
lie  in  a  —  b,  a. 

national  Limits 

40.  In  the  next  chapter  we  shall  have  a  good  deal  to  say  of 
infinite  sequences  and  their  limits.  We  proceed  to  define  them 
as  far  as  rational  numbers  are  concerned. 

Let  ^  be  a  set  of  rational  numbers  such  that : 

1°.  It  is  determined  whether  a  given  number  belongs  to  A 
or  not. 

2°.  There  is  a  first  number  a^,  of  the  set,  a  second  number  a^; 
and  in  general,  after  each  a„  follows  a  certain  number  a^+i- 

The  set  A  is  then  called  an  infinite  sequence  or  simply  a  sequence, 
and  is  denoted  by 

A==a^,  ag,  •••  or  by  A  =  la„\. 


RATIONAL   LIMITS  25 

EXAMPLES 

41.  1.    K  we  take  ««  =  n, 

A  ^1,2,  3,  ...  =  {»} 
is  a  sequence. 

2.  If  we  take  a„  =  - ,  we  get  a  sequence 

^  =  1,1,  i,...  =  {U. 

2    3  l«/ 

3.  If  an—  1,  we  get  a  sequence 

^  =  1,  1,  1,  1,  ...={1}. 

4.  Let  vl  consist  of  the  rational  numbers  lying  in  the  interval  0,  1,  arranged  in 
order  of  magnitude. 

This  set  A  is  not  a  sequence.  For,  although  it  is  perfectly  determined  what 
numbers  belong  to  A,  and  although  there  is  a  first  element  ui  =  0,  there  is  no  second 
element,  no  third  element,  etc.,  by  33,  2.  Thus,  while  condition  1°  is  satisfied, 
condition  2°  is  not. 

42.  1.    We  define  now  the  term  limit. 

Let  Z  be  a  fixed  rational  number.  We  say  :  I  is  the  limit  of  the 
sequence  J.  =  {a„|,  when  for  each  positive  rational  number  e,  small 
at  pleasure,  there  exists  an  index  m,  such  that 

\l-a,\<e  (1 

for  every  n>m. 

In  symbols  we  write  ,     ,. 

■^  1  =  lim  a^ ; 

n=oo 

we  also  use  the  shorter  forms 

Z  =  lim  a^,  or   a^  =  I, 

when  no  confusion  can  arise. 

We  shall  also  employ  at  times  the  symbol 

I  =  lim  a„. 

A 

When  I  is  the  limit  of  A,  we  say  A  is  a  convergent  sequence, 
and  that  a„  converges  to  Z  as  a  limit. 

2.  Notation.  We  shall  find  it  extremely  convenient  to  employ 
the  following  abbreviation : 

€>0,    m,    |Z— a„|<e,    w>w  (2 

to  mean  that,  for  each  positive  rational  e  there  exists  an  index  m, 
such  that  I Z  —  a„|  <  efor  every  n'p-m. 

The  reader  should  therefore  repeat  the  italics  often  enough  to 
himself  to  be  able  to  read  the  line  of  symbols  2)  without  hesitation. 


26 


RATIONAL  NUMBERS 


3.    The  reader  should  observe  that  from  2)  we  can  conclude 
also  that  for  each  positive  M  there  exists  an  m\  such  that 


i'-«.i<^' 


If,  therefore,  {a„|  has  the  rational  limit  Z,  we  can  write 
€>0,   m,   K-«»|<-^^    n>m. 


(3 


We  have,  of  course,  changed  the  notation  slightly  in  3)   by 
dropping  the  accent  of  m'. 

43.    The  graphical  interpretation  of  this  definition  will  prove 
most  helpful  in  our  subsequent  reasoning. 


Ui 


Let  us  lay  off  the  points  on  our  axis,  corresponding  to  the  num- 
bers «„,  also  the  point  corresponding  to  I.  On  either  side  of  I  lay 
off  the  points  I  —  e^  I  -\-  e.  These  determine  an  interval,  marked 
heavy  in  the  figure,  which  we  shall  call  the  e-interval. 

If  now  I  is  the  limit  of  the  sequence  A,  there  must  exist  for 
each  little  e-interval,  an  index  m,  such  that  the  images  of  all  the 
numbers  a^+i?  ^m+21  "•  f^'H  within  the  e-interval.     See  39. 

In  general,  as  e  is  taken  smaller  and  smaller,  the  index  w 
increases.  The  definition,  however,  only  requires  that  for  each 
given  e  there  exists  some  corresponding  m  such  that  42, 1)  holds 
for  every  n  greater  than  this  m. 

44.  Another  useful  graphical  interpretation  of  the  definition  of 
a  limit  is  the  following. 


H  'o.* 


RATIONAL   LIMITS  27 

We  take  two  axes  a:,  y  as  in  analytic  geometry.  On  the  a;-axis 
mark  off  points  1,  2,  3,  •••  at  equal  distances  apart.  Lay  off 
the  numbers  a^,  a^^  a^,  •••  as  ordinates  on  lines  through  the 
points  1,  2,  3,  •••  parallel  to  the  ?/-axis.  (See  Fig.)  These 
points  we  may  consider  as  the  images  of  the  numbers  a„.  On 
either  side  of  the  line  y  =  I,  draw  parallel  lines  at  a  distance  e 
from  it.  We  get  then  a  band,  shaded  in  the  figure,  which  we 
shall  call  the  e-hand. 

Then,  if  I  is  the  limit  of  A,  there  exists  for  each  e  an  index  m, 
such  that  the  images  of  all  the  numbers  a,„+i,  «m+2?  •••  fall 
within  the-corresponding  e-band. 

45.  EXAMPLES 

1.  ^  =  -[-]■;    lim  n!„  =  lim-  =  0. 

L  71 J  n 


2. 


M^-D-'H'-i) 


3.  ^  =  1,  -i,  +i    -  1,  •••  «„=  (-l)»+i-;  lima„  =  0. 

n 

The  reader  will  find  it  helpful  to  construct  the  graphs,  ex- 
plained in  43,  44,  for  each  of  these  sequences. 

46.  If  it  is  known  of  tivo  rational  numbers p^  q^  that  \p  —  q\<  €, 
however  small  e  >  0  may  he  taken,  then  p  =  q- 

For,  if  p  4^  q,  say  p  >  q,  then  p  —  q  is  a  definite  positive 
rational  number  ;  call  it  d.  Then  \p  —  q\  is  not  <  d,  and  this 
contradicts  the  hypothesis.     Hence  p  =  q- 

47.  A  rational  sequence  A  =  \a„l  cannot  have  two  rational  limits  Z,  V . 
For,  since  a^  =  I,  we  have  by  definition, 

e > 0,  Wj,  \l  —  a„\<€,  n> m^.  (1 
Also,  since  a„  =  T,  we  have 

€>0,  m^,  I?'  — a„|<e,  n>m^.  (2 
Let  m>mp  m^;  then  from  1),  2)  follows 

e>0,    ?w,    |Z— a„|<e,    w>m.  (3 

e>0,    w,    1^'  — a„|<e,   n>m.  (4 


28  RATIONAL   NUMBERS 

The  inequalities  3),  4),  holding  now  for  the  same  m,  we  can  add 

them,  and  get,  by  38,  3), 

\l-l'\<2e.  (5 

But  since  e  is  small  at  pleasure,  so  is  2  e.      The  inequality  5) 
gives,  by  46,  ^^^,^ 

48.  If  the  rational  sequence  ]aj<,  has  a  rational  limit  I,  there  exists 

an  index  m,  such  that 

b<a„<  c;       n>m,  (1 

where  h  is  any  rational  number  <  Z,  and  c  any  rational  number  >  I. 
For,  since  a„  =  Z, 

e>0,    m,    |Z  — a„|<e.        n>m. 

.-.  l-€<a„<l  +  e.  (2 

Since  e  is  arbitrarily  small,  we  can  take  it  so  small  that 

l  —  e>b,    l  +  e<c. 
Then  2)  gives  1). 

49.  Let  the  two  sequences  la„|,  \b^l  have  the  rational  limits  a,  b 
respectively.      Then 

lim  (a^  +  5„)  =  a  +  b;    lim  (a„  —  5„)  =a  —  b. 
For,  l(a  +  5)  -  («„  +  J„)  I  =  |(a  -  a„)  +  (^  -  6„)| 

<\a-a^\  +  \b-b^\  (1 

by  37,  3). 

Since  a„  =  a,  we  have 

e>0,    m',    |a— a„l<e/2.       n>m'.  (2 

Since  5„  =  6,  we  have 

e>0,    m",    \b-b„\<€/2.       n>m".  (3 

By  choosing  m  so  large  that  m>m',  m'\  we  can  suppose  2),  3) 
hold  for  the  same  m.* 

*When  a>h,  a'^c,  a^d- 
vf^  shaW  oit&n  set  vaoTQ  sh.ovtlY  a>6,  c,  d"« 

Similarly  a^O,  b^O,  c^O-- 

may  be  written  more  shortly  a,  b,  c,  •••  ^  0 


RATIONAL   LIMITS  29 

Then  1)  becomes,  using  2),  3), 

I  (a  +  5)  -  (a„  +  6„)  I  <  I  + 1  =  e.        n>m. 

This  states  that 

lira  (a„  +  h„^=a  +  h. 

Similarly,  we  prove  the  other  half  of  our  theorem. 

50.    If  the  tivo  sequences  ja^j,  \hj<^  have  the  rational  limits  a,  h 
respectively,  then 

lim  a„5„  =  ah.  {X 

For,  I 

^^  =  a5  -  a„6„  =  a(5  -  6„)  +  5„(a  -  a„). 

.-.  \d,\<\a\\b-hr,\^\br\\a-a^\,  (2 

by  37,  3),  5). 

Since  h^  =  6,  we  have,  by  48, 


\K\<B. 

n  >  m'. 

Also,  by  42,  3, 

i^-*"|<2i«r 

W>7?i". 

Since  a„  =  a,  we  have,  by  42,  3, 

k        ^n\<^^' 

n  >  m'". 

Evidently  by  taking  m  large  enough,  we  can  use  the  same  m  in 
these  three  inequalities. 
Then  they  give  in  2) 

which  proves  1). 

51.   Let  the  two  sequences  \a„},  {5„|,  have  the  rational  limits  a,  5, 
iespeetively .     Let  h  and  b„^0. 

Then 

lim^-  =  ?.  (1 

o„     b 


30  RATIONAL   NUMBERS 

For, 

d  ^  ^      «n  ^  «^»  -  ^^«  —  (^^»  —  ab^  +  {ah  -  a„b') 

^a(b„-b)  ^  a-a„^ 
bb^  b^ 

Since  Jt^ 0,  |J|  >0.     Let  J5  be  a  rational  number,  such  that 

Then,  by  48,  there  exists  an  m,  such  that 

\b^\>B.         71  >m.  (3 

Also,  by  42,  3, 

e  >  0,  w,  1  ^  —  ^„  I  <  -pr-' —  n>m.  (4 


e>0,  m,  |a— a„|<  — .         n>m.  (5 

By  taking  w  large  enough,  we  can  use  the  same  m  in  these 
inequalities.     Putting  3)  in  2),  we  get 

<|  +  |  =  ^'  by  4),  5), 


2      2 
which  proves  1). 


CHAPTER   II 

IRRATIONAL  NUMBERS 

Insufficiency  of  R 

52.  Although  the  system  of  rational  numbers  R  is  dense,  and 
so  apparently  complete^  it  is  easy  to  show  that  it  is  quite  insufficient 
for  the  needs  of  even  elementary  mathematics. 

Consider,  for  example,  the  length  h  of  the  diagonal  of  a  unit 
square.     This  length  is  defined  by  the  equation 

83  =  2.  (1 

We  can  show  there  is  no  number  in  R  which  satisfies  1).  For, 
suppose 

where  a,  b  are  two  positive  integers,  which  we  can  take  without 
loss  of  generality,  relatively  prime. 
Then  1)  gives 

a2=2  52. 

Let  jt)  be  any  prime  factor  of  h.  It  is  then  a  divisor  of  a\  and 
so  of  a.  Thus  a  and  b  are  both  divisible  by  p.  They  are  thus 
not  relatively  prime,  unless  p  =  l. 

Thus  5  =  1;  and  8  is  an  integer.  But  obviously  there  is  no 
integer  whose  square  is  2. 

53.  1.  A  similar  reasoning  shows  that 

■\/a 

does  not  lie  in  R,  unless  a  is  the  nth.  power  of  a  rational  number. 

31 


32  IRRATIONAL   NUMBERS 

The  numbers 

6  =  2.71828-,     7r  =  3.14159. 

can  be  shown  to  be  irrational ;  the  numbers 

log  a;,     e*,     sin  a;,     tana; 

are  in  general  not  rational. 

2.  Let  us  show  that  ,     ,       r 

I  =  log  5, 

the  base  being  10,  does  not  lie  in  R. 
If  I  were  rational,  we  should  have 


1  = 

a 

"V 

where  a,  h  are  integers. 

Then                               a 

10*  = 

5.     . 

•.  : 

10"  =  5^  (1 

Obviously  I  cannot  be  negative  ;  we  can  thus  suppose  a,  5  >  0. 

Now  any  integral  positive  power  of  10  is  an  integer  ending  in 
0  ;  while  any  integral  positive  power  of  5  ends  in  5. 

Thus  1)  requires  that  a  number  ending  in  0  should  equal  a 
number  ending  in  5,  which  is  absurd.     Hence  I  is  not  rational. 

Cantor's  TJieory 

54.  1.  The  preceding  remarks  show  clearly  the  necessity  of 
forming  a  more  comprehensive  system  of  numbers  than  R.  How 
this  may  be  done  in  various  ways  has  been  shown  by  Weierstrass, 
Cantor,  Dedekind,  Hilbert,  and  others. 

We  adduce  now  certain  considerations  which  lead  up  to  Cantor's 
theory. 

We  have  seen  no  rational  number  exists  which  satisfies  the 

equation 

a;2  =  2.  (1 

It  is,  however,  possible  to  determine  an  infinite  sequence  of 
rational  numbers 

such  that  ,.        or, 

lim  a  J  =  2. 


CANTOR'S   THEORY  33 

The  method  we  now  give  for  finding  such  a  sequence  A  has  no 
practical  value ;  it  has,  however,  theoretical  importance. 
For  aj,  we  take  the  greatest  integer,  such  that 

In  the  present  case,  a^  =  l. 
From  the  numbers 

we  take  for  a^  the  number  whose  square  is  <  2,  while  the  next 
number  of  2)  gives  a  square  >  2. 
Suppose  ^ 

Then  1^ 

S'<2<(a,  +  A)2. 
-brom  the  numbers 

we  take  for  a^  the  number  whose  square  is  <  2,  while  the  next 
number  of  3)  gives  a  square  >  2. 
Suppose  ^^ 

3        2^102 


Then  /  i  \2 

2^9^  I  n     _1__L  )    . 

i02y 


«3'<2<h3  + 


We  may  proceed  in  this  way  without  end,  and  get  thus  an 
infinite  sequence  of  rational  numbers, 


a„  =  a-,  -\ — *^  : 
2        1^10 


a„  =  a,  +  -^  =  a,  -}-  fi  +  -^ 
3        2-r-^^2        i^lO^K 


102 


4        3  ^  103        1  ^  10      102  ^  103 


a    —  n         A-     ""-1    —  //     4-  iiil  -U    "2    _i_  , . .  4_    "ra-1 
«„-«„-!+   ^^„_,-«l+jQ  +  J^+  +10^ 


34  IRRATIONAL   NUMBERS 

By  actual  calculation  we  find  the  numbers 

rtj,       flfg,       «3,       a^, 
are  respectively 

1,       1.4,       1.41,       1.414,       1.4142, 

2.   We  show  now  that 

lim  a„2  =  2. 
For,  from 

/  1      \2 

a2<2<    «„+  ^ 


10 
we  have 


•••|2-««^1<T7^+       ^ 


Obviously  now,  for  each  rational  e  >  0,  we  can  find  an  m,  such 

that 

3      ,        1       ^ 


Then 

1 2  —  a„2 1  <  e.         n>m. 
Hence 

lim  a„2  =  2. 

55.    1.   The  method  given  in  54  for  forming  the  sequence  a^, 
ag,  ag,  •••  admits  a  simple  graphical  interpretation. 

103= i-« 


0  1         1,41.5       2  2 

We  first  divide  the  indefinite  right  line  L  into  unit  segments ; 
flfj  is  end  point  of  one  of  these  segments.  In  the  present  case 
a^  =  1. 

We  next  divide  the  segment  1,  2  into  10  equal  parts ;  a^  is  the 
end  point  of  one  of  these  segments.     In  the  present  case  a^  =  lA. 

We  next  divide  the  segment  1.4,  1.5  into  10  equal  parts  ;  a^  is 
the  end  point  of  one  of  these  segments.  In  this  way,  we  continue 
subdividing  each  successive  little  interval  or  segment  into  10 
smaller  parts,  without  end. 


CANTOR'S   THEORY  35 

We    observe  that  each  little  segment  is  contained  in  the  im- 
mediately preceding  one,  and  therefore  in  all  preceding  ones. 
Also,  that  the  lengths  of  these  segments  form  a  sequence 


whose  limit  is  zero. 


1      1      J-      J- 

'      10'      102'      103' 


56.  1.  The  method  of  54  may  be  used  to  find  an  infinite  sequence 
of  rational  numbers 

which  more  and  more  nearly  satisfy  the  equation 

10^^  =  5, 
which  defines  log  5. 
We  find  : 

ftj  =  0,     a2=  .6,     a^=  .69,     a^=  .69S,     ••• 

2.  The  same  method  may  evidently  be  applied  to  any  problem 
which  defines  an  irrational  number.  In  each  case  it  leads  to  a 
sequence  of  rational  numbers 

fli,         (Zo*         ^31         •  *  *  -^ 

such  that 

1°.  Each  number  a„  satisfies  more  nearly  than  the  preceding 
ones  the  conditions  of  the  problem. 

2°.  For  each  positive  rational  e,  arbitrarily  small,  there  exists 
an  index  m,  such  that 

la„-aj<e, 
for  every  n,  v>m. 

57.  Regular  Sequences.  1.  It  is  this  second  property  of  the 
sequences  A,  that  '^antor  seizes  on  to  construct  the  elements  of 
his  number  system.     We  lay  down  now  the  following  definitions. 

Any  infinite  sequence  of  rational  numbers 

^\i     ^2^     ^^3'     *" 

which  ha""  property  2°  in  56  is  called  regular. 

As  in  42,  2,  we  shall  indicate  this  property  by  the  abbreviated 

notation :  „  ,  , 

e  >  0,     m,      a„  — a^  <€,     n,v>m.  (1 


36  IRRATIONAL  NUMBERS 

2.  Every  regular  sequence  defines  a  number^  which  we  represent 
by  the  symbol 

«  =  («!,    «2'    ^3'    ••■)• 

The  totality  of  such  numbers  forms  a  number  system^  called  the 
system  of  real  numbers. 

We  shall  denote  it  by  9fJ,  which  may  be  read  German  R. 

For  the  convenience  of  the  reader,  we  shall  denote  in  this  chap- 
ter the  new  numbers,  i.e.  the  numbers  in  QfJ,  by  the  Greek  letters 
a,  /3,  7,  •••;  while  the  Latin  letters  a,  b,  c,  •••  denote  numbers 
in  the  old  system  M. 

To  see  if  a  given  sequence  is  regular,  we  must  see  if  the  in- 
equalities 1)  are  satisfied.  For  this  reason  we  shall  speak  of  these 
inequalities  as  the  e,w  test. 

3.  The  €,m  test  is  equivalent  to  the  following : 

e  >  0,     wi,     I  a^  —  a„,  I  <  e,     w  >  m.  (2 

The  difference  between  1),  2)  being  that  in  |a„  — a„,  |,  only  one 
index,  n,  varies. 

For,  when  1)  holds,  2)  is  satisfied.  For  we  pass  from  1)  to  2) 
by  setting  v  =  m  in  1). 

Conversely,  if  2)  holds,  1)  is  satisfied. 

For,  since  e  in  2)  is  small  at  pleasure, 
let  us  take 


p  >  m. 

Adding  the  inequalities,  we  get,  by  38,  3), 

l««  — «vl<°"»     n,v>m, 
which  is  1). 


-1 

Then  2)  gives 

|««- 

1  ^  o" 

Also 

\a,- 

-<«»1<2 

CANTOR'S   THEORY  37 

4.    We  observe  finally  that  we  may  replace  n,  v>m  in  1)  by 
w,  v>  m. 

For,  if  11^  /Q 

for  every  w,  i'  ^  w,  it  is  true  for  every  n^  v>  m.  Conversely,  if  3) 
is  true  for  every  n,  i^  >  m,  it  is  true  for  every  n,  v  ^  m  +  1.  We 
would  therefore  in  1)  replace  m  hy  m  +  1. 

58.  EXAMPLES 

1.   That  the  sequeuces  A,  defined  in  54,  are  regular,  is  readily  shown.     We  have 

,  Ui  ,         ,    a„-i 
a„  =  ai  +  —  +  •••  H — ^  • 
10  10"-i 

0,^  =  a\ +  —-+•■•  -\ 


10  10"-! 

For  simplicity,  suppose  v  >  « ; 

then  „,_„,  =  ii.  +  i|^+...+i=l,  <-!,,.  (1 

as  the  considerations  of  55  show. 
If  we  choose  m  so  large  that 

i0^i<^' 
then,  by  1), 

dv  —  ««<«•         n,  v'>m. 

The  €,m  test  is  therefore  satisfied. 

2.    Consider  the  sequence 


Here 

If  we  take  now 
then 


1     _1     1     _1 

1'        2'   3'       4'    "* 

|a„-a,|  =  |^±-b-+-.  (2 


\n       v\  ^  71       V 
2 


m> 


1111 

-  +  -< — n,  j'>m. 

n      V      m      m 


Hence  2)  gives 

3.    Consider  the  sequence 


|««-«^,|<|+|  =  e. 


1,   1,   1,   1,   1, 


Here  «^^  _  «^  ^  1  _  1  =  0, 

and  this  sequence  evidently  satisfies  the  e,ni  test,  and  is  therefore  regular. 


38  IRRATIONAL   NUMBERS 

4.  Consider  ^        i    i        i     . 

1,  —1,  1,   —1,  ••• 

^^^®  |an-a^,|  =  0  or  2. 

Evidently  no  m  exists,  such  that 

I  «„  —  a^  I  <  e.        n,  v>-m. 
The  sequence  is  thus  not  regular. 

5.  Consider  1    2    3    4    ••• 

and  the  e,m  test  is  obviously  not  satisfied.     The  sequence  is  therefore  not  regular. 

59.  For  any  regular  sequence  of  rational  numbers  a^,  ag,  •••  there 
exists  a  positive  number  M,  such  that 

\a„\<M.        »i=l,  2,  3,  ...  (1 

For,  the  sequence  being  regular, 

e  >  0,  w,  I  a„  —  «;„  I  <  e.         n>-m. 

Hence  a^-e<a„<a^  +  e.  (2 

Let  M  be  taken  greater  than  any  of  the  m  +  2  numbers. 

Then  2)  proves  1). 

60.  The  elements  of  9?  have  as  yet  no  arithmetic  properties ; 
these  we  proceed  now  to  assign,  employing  the  method  already 
used  in  the  systems  ^  and  M. 

Our  first  step  is  to  place  ^  in  relation  to  M. 

Let  a  =  (aj,  a^-,  •••) 

be  an  element  of  9^^.     If  there  exists  a  rational  number  t»,  such  that 

lim  a„  =  a, 
we  say  a  =  a. 

61.  1.   Every  number  a  of  R  lies  in  9i. 
For,  consider  the  sequence, 

a  +  h  «  +  i,  «  +  i»  - 


CANTOR'S   THEORY  39 

This  sequence  is  regular,  since 

1      1 

a„  —  a^  = 

n      V 

The  number         cc  =  (a  +  l,  a+|^,  a  +  J,  •••) 

therefore  lies  in  9?. 

On  the  other  hand,  a^  =  a. 
Hence  a  =  a. 

2.    Let  ap  a^^  •••  be  any  sequence  of  rational  numbers,  having  0 
as  limit ;  then 

0=(ai,  ^2,  ...). 

In  particular,  0  =  (1,  l,  i,  ...) 

—  r— 1    —1—1  ...^ 

—  ^^        ^1  2'  3'         J 

_ri  —  1  1  —  4  '-'^ 

=  (0,0,0,...). 

62.    1.   We  define  now  the  terms  equal,  greater  than,  less  than. 
The  object  of  this  is  simply  to  arrange  or  order  the  elements  of  9fJ. 
Let 

a=(fljp  ^2,  •••),  ^  =  (^1,  h^,  ...). 
We  say 

a  =  ^,  when  lim  (a„  —  5„)  =  0  ;  (1 

or,  what  is  the  same  thing,  when 

€>0,  m,     \a„  —  b„\<i€.        n^nfi.  (2 

2.  We  say  a  >  /S  when  there  exists  a  positive  rational  number 
r  and  an  index  w,  such  that 

ci'n  —  ^n^f-  n>m.                            (S 

We  say  similarly,  «  <  ;S,  if 

6„-a„>r,  w>w.                            (4 

or                                            a„  -  5„  <  —  r.  (5 

3.  Numbers  of  'Si  which  are  >  0  are  called  positive ;  those  <  0 
are  negative. 


40  IRRATIONAL   NUMBERS 

63.  It  can  be  shown  that  from  this  definition  of  equality  and 
inequality  the  usual  properties  of  these  terms  can  be  deduced. 

For  example, 

7/  «  =  /3,  /3  =  7,  then  a  =  y. 
For,  setting 

«  =  («!,  a^,  ...) 

we  have 

««  -  ^«  =  (««  -  ^«)  +  (^«  -  0  =  0, 
since 

by  hypothesis. 

64.  If  a  =  (aj,  a^,   ...)  =  (aj',  a^',   .-.),  we  say  (a^,  a,^,  •••)  and 
(aj',  ag''  •")  ^^^  different  representations  of  the  same  number  «. 

Every  number  a  in  ^  admits  an  infinity  of  representations. 

In  fact,  there  are  obviously  an  infinity  of  rational  sequences 

2j,  ^2'  ^31    '•* 

having  zero  as  limit. 

Then  ^      ,  ,  n 

represent  an  infinity  of  representations  of  «. 

65.  1.    We  wish  to  appl}-  the  definition  of  62  to  the  case  that 
one  of  the  members,  sa}^  /3,  is  a  rational  number  h. 

Let  a  =  b. 

For  yS  =  5,  we  can  take  the  representation 

^  =  5  =  (6,  6,...).  (1 

Then  62,  1)  requires  that 

lim  (a„  —  5)  =  0 ; 

whence  lim  a„  =  b. 

Thus  the  definitions  of  60  and  62  are  in  accord  for  this  case. 


CANTOR'S   THEORY  41 

2.  Let  a>h. 

Since  J„  =  ^  by  1),  the  relation  62,  3)  becomes  here 

a„  —  b>r,         n>m.  (2 

Let  a  <  b. 

Then 

h  —  a^>r,         n>m.  (3 

or 

a„-b<-r.  (4 

3.  If  cc=  (aj,  flfgi  "0'^  ^1  ^^^^^6  exists  an  index  m,  and  two  positive 
rational  numbers  A,  B,  such  that 

A<a^<B;         n>m.  (5 

and  conversely . 

For,  set  6  =  0,  then  2)  gives,  replacing  r  by  J., 

a„  >  J.  >  0.         n>m.  (6 

On  the  other  hand,  59  gives 

\a„\=an<B.         n>m.  (7 

From  6)  and  7),  we  have  5).     The  second  half  of  the  theorem 
is  obvious,  by  2. 

4.  Similarly,  we  have 

If  a=^(a^^  ag,  •••)<0,  there  exists  an  index  m,  and  two  negative 
rational  numbers  —A,  — -B,  such  that 

—  A<a,^<—B;         n>m. 

and  conversely. 

5.  From  3  and  4  we  have 

If  a  =  («!,  ^2'  **')t^^5  the7'e  exists  an  index  m,  and  two  positive 
numbers  A,  B,  such  that 

A<\a„\<B;         n>m. 

6.  In  any  number  a  =  (^a^,  a^,  '■•')^0,  the  constituents  a„  finally 
have  one  sign. 

This  follows  at  once  from  3  and  4. 


42  IRRATIONAL   NUMBERS 

66.  1.    Let  a  =  (aj,  a^^  •••). 

//  a„>a,         n>m.  (1 

Then  a^a.  (2 

For,  suppose  a<a.     Then,  by  65,  2,  there  exists  an  r>0,  and 

an  m,  such  that 

a  —  an>r.         n>m. 

Hence 

a  >  a„  +  r  >  a^ ; 
and  therefore 

a„  <  «, 

which  contradicts  1).     Hence  2)  holds. 

2.    Similarly,  we  show : 

Let  a  =  (<Xj,  ^21  •••)• 

if  a^^a,         n>m, 

then  a<a. 

67.  1.    If  from  the  sequence 

1^  2'  '\^      ***  V 

which  defines  the  number  a,  we  j?^cZ:;  ow^  a  sequence 

a,^,    a,^,    «,3,    ...  (2 

M^ Aere  t J  <  ^2  <  ^3  "  • ;   then  also 

2%e  sequence  2)  ^s  regular.     For,  since  1)  is  regular, 

e>0,    m,    |a^— a^|<e,    n^v>m. 
But  then 

I  ^''i^  —  ^ij  <  f  1    r,  s  <  «,    ix  >  m. 

Hence  2)  is  regular,  and  defines  a  number  /3. 

We  show  now  a=  ^.     Since  2)  contains  only  a  part  of  1), 

t„^/i,    n=\,  2,  3,  ••• 
Since  1)  is  regular, 

\a„  —  a,  \<e.         n>m. 

Hence,  by  62,  1,  «  =  yS. 


CANTOR'S   THEORY  43 

2.  As  corollary  we  have  : 

The  number  «=  (a^  a^^  •••)  is  not  altered,  if  we  remove  from  or 
add  to  the  numbers  in  the  parenthesis,  a  finite  number  of  rational 
numbers. 

3.  We  have  also  : 

If  in  «=  («!,  fljg,  ...),   yS=  (b^,  ^2'  •") 

<^n=K^     n>m; 
then  a  =  /S. 

68.  1.  If  (t=  (^j,  ^21  *••)  =5^  0,  ^Aerg  cannot  be  an  infinite  number 
of  constituents  a^=Q. 

For,  say 

Then,  by  67,  1, 

But 

(a,„a,^,  ...)  =  (0,  0,  ...)=o. 

Hence  a  =  0,  which  is  a  contradiction. 

2.  If  a  ^0,  we  can  choose  a  representation  (a^,  ag,  •••),  ^w  which 
all  the  a„^0. 

For,  let  /    r        r         \  /^t 

be  any  representation  of  «.  By  1,  it  contains  but  a  finite  number 
of  zero.  If  we  leave  these  zeros  out  of  1),  we  do  not  change  the 
value  of  «,  by  67,  2 ;  and  get  thereby  a  representation  of  «,  none 
of  whose  constituents  are  zero. 

69.  1.  Having  ordered  the  elements  of  ^,  we  proceed  to  define 
the  rational  operations  upon  them. 

Addition. 
be  two  elements  of  9J,  different  or  not ;  then 

«  +  /3  =  («!  +  &!,  ^2  + ^2'   •••)•  G 


44  IRRATIONAL   NUMBERS 

To  justify  this  definition  of  addition,  we  show  first  that 

is  a  regular  sequence. 

Since  a^,  a^,  •••  is  a,  regular  sequence,  we  have 

e  >  0,    m,    I  a„  —  a^  I  <  e/2,    n,  v>m.  (3 

Since  6j,  b^,  •••  is  regular,  we  have 

e<0,    m,    |5„-5,l<e/2,    n,v>m.  (4 

Evidently  we  can  take  m  so  large  that  3),  4)  hold  for  the 
same  m. 
Now 

I  («n  +  ^n)  -  Q^v  +  ^.)  I  =  I  {an  -  «.)  +  Q^n  "  ^v)  I 

<  |a„  -  a,|  +  16^  -  6,|,  by  37,  3)  ; 
<|  +  |=6,by3),4). 

Thus  2)  is  regular,  and  defines  a  number. 

2.    We  show  next  that,  if  «,  /S  are  rational  numbers,  say  «  =  a, 
/3  =  5  ;  then  a  +  yS,  as  defined  by  1),  is  a  +  6. 
Since  a  is  a  rational  number  a, 

lim  a„  =  a,  by  60. 
Similarly,  lini6„  =  &. 

lim  (a„  +  ^,)  =  lim  a„  +  lim  6^  =  a  +  5,  by  49. 
Thus  by  60,        ,      ,  ,  ,    ,  .  ,    , 

Hence  by  1),  ,    o  .   a 

70.    1.   If  /3>y,  then  a  +  ^>a  +  y. 

Let  7  =  (cp  (?2'  "■)■    Since  /3>  7,  there  exists,  by  62,  2,  a  positive 
rational  number  r,  such  that 

^i  >(^n  +  r.         n>  m. 
Hence,  adding  a„, 

dn  +  K  >  «n  +  Cn  +  ^' 

Hence,  by  62,  2,  „ 


CANTOR'S   THEORY  45 

2.    From  1,  we  conclude,  as  in  26,  that 
Tfa-\-^=a  +  j,  then  y8  =  7. 

71.    1.  Subtraction. 

This  is  the  inverse  of  addition ;  we  define  it  as  we  did  in  %  and 
M,  viz. :  The  result  of  subtracting  ^  from  a  is  the  number  or  num- 
bers I,  in  9?,  which  satisfy 

«=/3  +  ^  (1 

There  is  at  most  one  number  ^. 

For,  suppose  a  —  ^  +  7].  (2 

Then  1),  2)  give,  by  63, 

y8  +  |  =  /3  +  7;. 

Hence,  by  70,  2,  7;  =  |. 

To  show  that  1)  admits  one  solution,  we  prove  just  as  in  69,  1, 

that  ,  J. 

«i  —  ^i,  a^  —  b^,  ••• 

is  a  regular  sequence,  and  thus  defines  a  number 

If  we  put  this  value  of  |  in  1),  the  equation  is  satisfied. 
For,  ^  +  I  =  (^1,  b^j,  •••)  +  («!  -  Jj,  a^-  b^,  •••) 

=  (&i  +  flj  -  ^1,  63  +  «2  -  ^2'  •••)^  by  69,  1) 

=  («!,  flg,  •••)  =  «. 

2.  Thus  subtraction   is  always  possible  in  9?,  and  is  unique. 

The  result  of  subtracting  /3  from  a  we  represent  by  a  —  /3 ;    we 

have  then  n     r         r  7         n 

«  -  P  =  («i  -  Oj,  ^2  -  62,  •••)• 

3.  We  represent  0  —  «  by  —  a. 

Evidently,  ,  . 

—  «=(-«!,  —«2'  —^3'  •••)• 

We  observe  that         «+(—«)  =  0 ; 

a+(-/3)  =  «-/3; 
-(-«)  =  «• 


46  IRRATIONAL  NUMBERS 

72.  1.  -Z/«  is  positive^  —  «  is  negative;  and  if  a  is  negative,  —  a 
is  positive. 

For,  if  cc=  (aj,  a^,  •••)  >0,  we  have,  by  65,  3, 

a,i>A>0.         n>m. 
Now 

_  «=  (- a^,  -  ^2,  •••)'  i^y  "^ii  3. 

Hence,  by  1), 

—  a„<  — yl<0.         n>m. 
Hence,  by  65,  4, 

-  a  <  0, 

which  proves  the  first  part  of  the  theorem.      The  second  part  is 
proved  similarly. 

2.  All  the  numbers  of  dl  ^0  are  of  the  form  a  or  —  «,  where  a  is 
a  positive  tiumber. 

Let  /3  be  a  number  ^  0.  We  need  to  consider  only  the  case 
that  /3  is  negative. 

and  by  1),  —  yS  is  positive. 

73.  1.    Multiplication. 

The  product  of  a  by  /3  we  define  by 

a^  =  (a^b^,  a^b^,   •••)•  (1 

We  have  to  show  that 

is  a  regular  sequence. 

Let  e  be  a  positive  rational  number,  small  at  pleasure. 
Then,  by  59,  there  exists  a  positive  M.,  such  that, 

|a„|,     \b„\<M.         n>m.  (3 

Also,  since  the  sequences  |«„(,  \bn\  are  regular,  we  can  suppose 
m  in  3)  is  taken  so  large  that 

1««-«J,    |5„-6J<^.         n,  v>m.  (4 


CANTOR'S.  THEORY  47 

Now 

d^  =  aj)^  -  aj)^  =  a^h^  -  5  J  +  hX^n  -  « J- 

.-.  |(^„|^|a„||6„-6J  +  |5J|a„-aJ,  by  37, 

and  2)  is  regular. 

2.  If  a,  y8  are  rational,  say  «=(x,  /3=J,  we  show  that  ayS  as 
defined  in  1)  is  ah. 

For,  since  a  and  ^  are  rational, 

lim  a„  =  «,  lim  5„  =  5,  by  60. 
But  then,  by  50, 

lim  ajbn  —  lira  a„  lim  5„  =  a5, 
which  states  that  a^=  ah. 

74.  1.  The  formal  laws  for  addition  and  multiplication  are 
readily  proved.  We  illustrate  this  by  establishing  the  associative 
law  of  multiplication. 

We  wish  to  show  that 

a  .  ^7  =  a/3  .  ry.  (1 

We  have,  by  73,  1), 

Hence 

«./37  =  (aj,  ^2,   ••')(h^c^,  h^c^,   .••) 

Similarly, 

«/3  •  7  =  (^1*1  •  (^p    «2^2  '  ^2'     •••)•  (3 

Since  multiplication  is  associative  in  i2,  the  two  numbers  repre- 
sented by  2),  3)  are  identical,  which  proves  1). 

2.  As  a  consequence  of  the  associative  law,  we  have,  m,  n  being 
positive  integers, 

which  expresses  the  addition  theorem  for  integral  positive  exponents. 


48  IRRATIONAL   NUMBERS 

75.    1.   The  properties  of  products,  relating  to  greater  than,  less 
than,  are  readily  established  for  numbers  in  9?. 

If  a>  13,  and  7  >0,  then  ay  > /3y. 

For,  since  a>yS,  we  have,  by  62,  2, 

Since  7  >  0,  there  exists  a  positive  rational  number  c,  by  Qb,  3, 

such  that 

Cj^>c.         n>m. 

By  taking  m  sufficiently  large,  we  may  take  the  same  m  in  both 
these  inequalities. 
They  give 

««<?n  -  ^nCn  >Cr>0. 

Then,  by  62,  2, 

ay  >  /37. 

2.  From  1  follows : 
lfa>^>0, 

then  a"  >  /3".      w  positive  integer. 

3.  From  2  we  conclude  : 

If  a,  /3  >  0,  and  a"-  =  y8%  w  being  a  positive  integer,  then 

a  =  13. 

4.  IfO<a<l,thena''<a. 
For,  from 


we  have 
Also 
Hence 
Hence,  in  general, 


«2  <  «. 
a^  <  a^^ 
a^<  «. 
«"<  a. 


76.  1.  i^wZe  of  signs:  The  product  of  two  positive  or  two  negative 
numbers  in  9?  is  positive.  The  product  of  a  positive  and  a  negative 
number  is  negative. 


CANTOK'S  THEORY  49 

Let  «  >  0,  yS  >  0 ;  then  a/3  >  0. 

B}^  65,  3,  there  exist  two  positive  numbers  A,  B,  and  an  index 

m,  such  that 

a„>^,         K>^-         n>m. 
Hence 

aj)^  >AB>0. 
Thus 

a/3=:(ajb.^,  a^b^,  •••)  >0,  by  66. 

Ze^  «  >  0,  /3  <  0 ;  ^Agw  aj3<0. 
For,  by  65,  3,  4, 

a„>A,         h„<  —  B.         n>m. 

.'.  aJ)„<-AB<0. 
Thus,  by  66, 

«/3<0. 

In  a  precisely  similar  manner,  we  can  treat  the  other  cases. 

77.  1.    TJie  product  of  any  two  numbers  in  9?  vanishes  when,  and 
only  when,  one  of  the  factors  is  zero. 

In  the  product  a/3,  suppose  «  =  0  ;  then  «/8  =  0. 

Then 

«  =  (0,  0,  0,  ...),     ^  =  ib„b^,-). 

Conversely,  if  «/S  =  0,  either  «  or  /3  =  0. 
This  is  proved,  as  in  25. 

2.    If  a^  0,  and  «/3  =  a<^,  then  /S  =  7. 
Proof  same  as  that  for  26,  3). 

78.  1.  Division. 

The  quotient  of  «  by  ^  is  the  number  or  numbers  f ,  in  9^,  which 
satisfy 

«  =  /3f  (1 

There  are  two  cases,  according  as  ^=0,  or  ^  0. 


50  IRRATIOXAL  NUMBERS 

Case  I;  /3^0. 

Since  y8  ^fe  0,  we  may  suppose,  by  68,  2,  that  in 

all  6„  ^  0.     To  find  a  solution  of  1),  consider  the  sequence 
Zi^  is  regular.     For, 

^        ^  ^  _  ^  ^  <^rf>v  -  ^v^n  _  ««(^,.  -  ^n)  -  ^«(«^  —  «n). 


(2 


Hence 

By  59, 

\a^\<M.  n>m.  (4 

By  65,  5,  we  have 

A<\hj^\<B.  n>m.  (5 

By  taking  m  sufficiently  large  we  may  suppose  it  to  have  the 
same  value  in  4),  5). 
Then  4),  5)  gives  in  3), 

Since  the  sequences  {a„|,  \h^\  are  regular,  we  may  now  suppose 
m  taken  so  large  that  also 

Then  6)  gives 

Since  2)  is  regular,  it  defines  a  number 

=  (a^,  Og,  •••)  =  «,  by  73,  1), 
5  satisfies  1). 

That  this  is  the  only  solution  of  1)  follows  as  in  30, 1,  from  77,  2. 


Since 


CANTOR'S  THEORY  51 

Case  II;  0=0. 

We  can  reason  precisely  as  we  did  in  30,  2.  Hence,  when  the 
divisor  /3  =  0,  division  is  either  impossible  or  entirely  indetermi- 
nate.    For  this  reason  division  by  0  is  excluded. 

2.  We  have  thus  this  result :  in  the  system  9^?,  division  is  always 
possible  and  unique,  except  when  the  divisor  is  0,  when  division  is 
not  permissible. 

3.  The  result  of  dividing  «  by  yS,  we  represent  by  a/0  and  have 
therefore 


a      I  a^    a,^ 

^'""«  1= (1,1,1,-), 


0  Kh'h' 

This  is  called  the  reciprocal  of  0. 


■} 


79.  1.  The  system  9?  is  now  completely  defined  ;  its  elements 
have  been  ordered,  and  the  four  rational  operations  upon  them 
have  been  defined.  As  a  perfect  analogy  exists  between  the  sys- 
tems H  and  9?,  we  are  justified  in  calling  the  elements  of  9?  num- 
bers. In  the  future,  when  speaking  of  numbers,  without  further 
predicate,  we  shall  mean  the  numbers  m  9?.  As  already  stated, 
they  are  called  real  numbers. 

2.  In  the  e,m  test,  given  in  57,  we  were  obliged  at  that  stage 
to  take  e  rational.  This  is  now  quite  unnecessary,  and  we  shall 
therefore,  in  the  future,  suppose  e  is  any  positive  number  in  9?, 
small  at  pleasure. 

80.  1.  We  have  shown  in  61  that  9?  contains  all  the  numbers 
of  M ;  but  we  have  not  shown  that  it  contains  other  numbers. 

To  this  end,  we  show  that  there  is  a  number  a  which  satisfies 

x^  =  2.  (1 

This   is   easily  done.     For   in   54   we   determined   a   rational 

sequence  «^  =  1,  a^=  1.4,  a3  =  1.41,  ^..  -2 

such  that  ,  •        9     o  ^o 

iim  a„^  =  z.  (o 


52  IRRATIONAL   NUMBERS 

The  sequence  2)  is  regular  by  58,  1. 
Hence  ,  . 

is  a  number  in  9^. 


But,  by  73,  1), 


«' 


2  ///    2        /y    2 


{a^,  «2  '   •••)• 


Hence  3)  shows,  by  60,  that 

a2  =  2. 
Hence  «  is  a  solution  of  1). 

2.  As  we  saw  in  52  that  a  is  not  rational,  we  have  shown  there 
is  at  least  one  number  in  9?  not  in  R. 

But  the  reasoning  we  have  just  applied  to  V2  applies  equally 
to  "v^a,  when  this  latter  is  not  rational.  There  are  thus  an  infinity 
of  numbers  in  9?  not  in  R. 

Some  Properties  of  9? 

81.  If  a>  0,  there  are  an  infinity  of  positive  rational  numbers 
<  a,  and  also  an  infinite/  of  rational  numbers  >  a. 

If  a  is  rational,  the  theorem  is  obviously  true  by  33. 

Let  ,  . 

a  —  {a^,  a,,   •••). 

Then,  by  65,  3, 

0  <  J.  <  a„  <  ^.         w  >  w*  (1 

But  from  . 

a„>A, 

we  have,  by  Q6,  1,  _ 

a^A. 

Since  there  are  an  infinity  of  rational  numbers  between  0  and 
the  positive  rational  number  A,  the  first  half  of  the  theorem  is 
established. 

Using  the  other  part  of  the  inequality  1),  we  prove  similarly 
the  rest  of  the  theorem. 

82.  Between  «,  y8,  lie  an  infinity  of  rational  numbers. 

For,  let  a  <  /8 ;  then,  by  81,  there  exist  positive  rational  num- 
bers A,  B,  d,  such  that 

A<a,  ^>/3,  d<^—a. 


SOME   PROPERTIES  OP  9t  53 

we  can,  by  34,  2,  determine  the  positive  integer  n  so  great  that 

I>  7 

~<d. 
n 

Then,  at  least  one  of  the  numbers 

^  +  -,    A  +  2-,  ...  ^  +  (,,-1)^ 
n  n  ^  n 

falls  between  «  and  /3. 

83.  1.    The  system  9?  is  Archimedian ;  i.e.  for  each  pair  of  posi- 
tive numbers  «</3  there  exists  a  positive  integer  n,  such  that  na^ ^. 

For,  by  81,  there  exist  positive  rational  numbers 

a<a,  b>^. 

Since  the  system  B.  is  Archimedian  [34,   1],  there  exists  an 
integer  w,  such  that  , 

But  ,       ,  ry 

na  >na,    and    o>  p. 

Hence  ^ 

na>  jd. 

2.    For  any  pair  of  positive  numbers  a  <  /S  there  exists  a  positive 

n  such  that  „ 

P 

-<a. 
n 
Proof,  as  in  34,  2. 

84.  Between  a  and  ^,  «  <  /3,  lie  an  infinity  of  irrational  numbers. 
That  irrational  numbers  exist,  we  have  shown  in  80. 

Let  i  be  an  irrational  number,  r  a  rational  number,  and  n  a  posi- 
tive integer. 

Then  J  =-•>  k  =  i-{-r 

n 
are  irrational. 

For,  if  y  were  rational,  i=nj  is  rational.     This  is  a  contradiction. 

Similarly,  if  k  were   rational,  i=k  —  r  is  rational,  which  is  a 

contradiction. 


54  IRRATIONAL   NUMBERS 

This  established,  suppose  first  that  «  is  rational  and  positive. 
Let  i  be  any  positive  irrational  number. 
Then,  by  83,  2,  we  can  take  7i  so  large  that 

-<^-oc. 
n 
But  then 

«<«  +  -<  /3 ; 
n 

and  a  -f-  - 

n 
is  irrational. 

Suppose  now  that  a  is  irrational  and  positive. 

By  81,  there  exists  a  positive  rational  number  r,  such  that 

0<r<^-a. 
Then 

a  <  «  4-  r  <  /3 ; 

and  a-\-r 

is  irrational. 

The  cases  when  a,  /3  are  one  or  both  negative  are  now  easily 
treated. 

85.    The  si/stem  ?tt  is  dense,  i.e.  between  any  two  numbers  of  9? 
lie  an  infinity  of  numbers. 

This  follows  at  once  from  82  or  84. 


Numerical  Values  and  Inequalities 
86.    We  have  seen,  72,  2,  that  any  number  a^Q  can  be  written 

where  a^  is  a  positive  number. 

We  define  now,  as  in  36,  the  numerical  or  absolute  value  of  a  is 
+  «Q,  and  denote  it  by 

Then  by  definition. 

We  set  also 

|0|  =  0. 


NUMERICAL   VALUES   AND   INEQUALITIES  55 

87.    1.   We  have  now  the  following  fundamental  relations  : 

|a|  =  |-a|;  (1 

|a-/3|  =  |/8-a|;  (2 

|«±/3|^|«|  +  |/3|;  (3 

|«±/3|^||«|-|^||;  (4 

|«/3|  =  |«|  •  |yS|;  (5 

/3^0.  (6 


2.  From 
follows 

and  conversely. 

3.  From 
follows 

or  if  A  =  B, 


«i±«2  •••  ±«,„|<  |«i|H l-|«m|;  (7 

|«1  •«2  —  «'n|  =  |«ll  •  l«2l  —  l«».|-  C8 

\a\<A, 
—  A  <a<A; 


\a-^\<A,    |yS-7|<5, 
\a-ry\<:A  +  B; 
|«— 7|<2^. 


4.  As  the  demonstration  of  these  relations  is  exactly  the  same 
as  in  37,  38,  we  do  not  need  to  repeat  it. 

5.  If  ive  know  of  two  numbers  a,  /8,  that  | «  —  yS|  <  e  however  small 

e>0  is  taken;  then  „ 

«=  p. 

The  demonstration  is  the  same  as  in  46. 
88.    If  «  =  (aj,  a^,  •••),  then 

Since  the  sequence  a^  a^,  ••• 

defines  a  number,  it  is  regular. 

Hence  ^  ,  , 

€  >  0,  m,     I  a„  —  a^  I  <  e.         n,  v>m. 


56  IRRATIONAL  NUMBERS 

From  this  we  conclude  that  the  sequence 


is  regular. 

For,  by  87,  4), 


(1 


I  «„  I  —  I  «„  I  I  <  I  a„  -—  a„ 


Hence  1)  defines  a  number. 

Tq  shoiv  that  /S  =  I «|. 
First  suppose  a  =  0» 
Then 


Hence 
Therefore 


lim  a„  =  0. 

lim  |a^|=  0. 
/3=  0,  and  y8  =  |«|. 


Suppose  « :^  0.     Then,  by  65,  6,  the  constituents  a^  of  a  are  of 
one  sign,  for  n>m. 

a„=\aA.         n>m. 


If  «>0, 
Hence 

If  «<0, 
Hence 


"n        I  ^n  I 


=  a  =  !«!,  by  67,  3. 

a„=  —  |a„|.         n>m. 

^^        V        1^1  I'  '"''         l^wih   *m+l'   '^«»+2i  "'J 

=  —  a,  by  67,  3 


89.    In  the  following  articles  we  give  certain  equalities  and 
inequalities  which  are  often  useful. 


Let  0  <  «<  1 ;  then 


1  +  a 
1 


>l+a. 


(1 
(2 


NUMERICAL   VALUES   AND   INEQUALITIES  57 

To  prove  1),  let  us  suppose  the  contrary,  viz.: 

1     =1 

1  +  a 
Clearing  of  fractions, 

l<l-«2,  or  a2<0, 

which  is  a  contradiction. 
Similarly,  we  may  prove  2). 

90.  1.   Xf^«i,«2,...,«,„>0awc?P,„=(l  +  «i)(l  +  «2)...(l  +  a,J. 

P,„>1 +(«!  +  ... +  0-         m>l. 

Pm  >  1  +  («i  H h  a„0  +  («i«2  +  «1«3  H ^-  «m-l«/«)-       Wl  >  2. 

In  fact, 

Pg  =  (1  +  ftj)  (1  +  «2)  =  1  +  («j  +  «2)  +  «i«2  >!+(«!  +  «2)  ; 

-^3  =  A(l  +  «3)  =  1  +  («i  +  «2  +  "a)  +  «l"2  +  "l^'^S  +  "2^3  +  "l"2"3 

>  1  +  (rtj  +  «2  +  "3)  +  "l«2  +  "l"3  +  "2^3 

>  1  +  («j  +  tta  +  ttg). 

In  this  way  we  can  continue. 

2.  Similarly,  we  can  prove  : 
Let  0  <  «j,  a2,  •••  «„j<  1,  and 

^„,  =  (l-«l)(l-«2)-(l  -«,„). 
<l-(«l  +  -"  +  «m)  +  («l«2  +  «l"3+'-  +  '^m-l«m)-    ^>2. 

91.  The  demonstration  of  the  following  identities  is  obvious : 

-1— =  l  +  «  +  «2 +  ...+  ««-! +-2!L.  (1 

1  —  «  1—  a 

« +  e  _  tf        /3e-  aS  • 


58  IRRATIONAL   NUMBERS 

92.    Let  \8^\,  \b^\<^S,  and /3  =^  0  ;  let 


P  = 


1. 


Let  e>0  5e  small  at  pleasure  ;  tve  can  take  S>0  so  small  that 

'    P=^  +  -.  \a\<e.  (1 


by  37,  (2 


For,  by  91,  2, 

^_  Si^-V. 

K^  +  ^) 

Hence 

s^M  +  ^11, 

'-|/3H!/3|-^I 

king  8  so  small  that 

|^|-8>0. 

Let 

I«l'  l/3l<^, 

1^1,  \^\-8>h. 

Then  2)  gives 

0"    < — ^• 

'     '         P 

Hence,  if  we  take 

«<f, 

3  have  1). 

2^ 

A>0. 


93.    Let  «p  «2,  •••  «„  he  n  arbitrary/  numbers. 
Let  ^i-/3„;  7i-7«>0. 


then 

For,  from 


^  <  7i«i  +  ---  +  7A  <  (nr_ 
~7AH ^Jn^n^ 


"l>i, ...,  '^>x, 


/3i  -  /e„ 

7i«i  H h  7«««  >^(^i7i  H ^  /3„7n), 

which  gives  the  first  half  of  1).      The  rest  of  1)  follows  similarly. 


we  have 
Adding, 


NUMERICAL  VALUES  AND   INEQUALITIES  69 

94.  1.   Let  «  >  ^  >  0  ;   and  n  >  1,  «  positive  integer. 

Then 

n{a  -  /S)/3"-i  <  a"  -  /3"  <  n{a  -  /3)a"-l.  (1 

For,  by  direct  multiplication,  we  verify 

««  _  ;3"  =  (a  _  /3) (a"-l  +  a»-2y3  ^  a"-3y32  _, ^  y3«-i ^^ _  ^2 

In  the  second  parenthesis,  replace  a  by  /3.     Then,  since  «  >  /3, 

a"  -  yS'^  >(«  -  /3)(yS«-i  +  /S"-'  +  •••  w  terms), 

or  an  —  ^""^^  n{a  —  ^}/3"-~'^, 

which  is  a  part  of  1). 

If  in  2)  we  replace  /S  by  «,  we  get  the  other  half  of  1). 

2.  In  1),  set  a  =  1  +  S,  S  >  0,  ;8  =  1,  we  get 

(1  +  8y  >  1  +  nS. 
If  we  set  «  =  1,  yS  =  1  —  5,  1)  gives 

(1  _  By  >l-n8. 
We  have  thus 

(1  +  «)"  >  1  +  na,  «  ^fc  0  and  y.  —  1,  n  positive  integer.       (3 

3.  We  observe  that  1)  can  be  written 

a"  >  y8"-i [/3  +  w(«  -  yS)] ,  (4 

/3"  >«"-![« -M(a-yS)].  (5 

95.  Let  «j  •••  a,j  be  any  n  numbers. 

^»"~ ^^ 

is  called  their  arithmetic  mean. 

Let  «j  •  •  •  a„  5e  positive,  and  P^  =  ^\  '  f'-^  " '  ^w 

Then  P„ < ^„",  unless  the  as  are  all  equal,  when  P„  =  A^. 

If  aj  =  a2  =  •••  =  «„,  vl„  =  «i  and  P„  =  ai". 

Hence  P„  =  ^/. 

Suppose  now  the  a's  are  not  all  equal. 


60  IRRATIONAL   NUMBERS 


a.=(^bjt^X-(^^J^]< 


"i  +  '^aY 


Let  n  =  2 
We  have 

Hence  i^2<^2'- 

Let  n  =  2"^. 

Since  the  a's  are  not  all  equal,  at  least  two  of  them,  say  «],  ag- 
are  unequal. 
Then 


<'.«.<(^J. 


and  /      , 


Hence 


«:W.<(H^J(H^J-  O 


On  the  other  hand,  applying  our  theorem  to 

«T  +  a-i       «3  +  «4 

2      '         2     ' 

we  have  ,  ,  /      ,        ,        ,  „ \2 

«1  +  «2  .  «.S+«4^  /«T  +  «^  +  «,s  +  «4Y.  /2 

2  2    ~V         4         y  "^ 

Hence  1)  and  2)  give  r>  ^  j  4 

In  the  same  way,  we  may  continue  for  any  power  of  2. 
Let  2'"-! < w< 2'".     Set  /i  =  2'%  2"'-n=v. 
We  have,  by  the  preceding  case. 

Then  3)  gives 


Set  here  a 


^^^...^a^  +  .A^J^^^^  (4 


A* 


Since 


—  —  Jiff 


Dividing  in  4)  by  A^,  we  get 


LIMITS  i61 

96.    From  algebra  we  have  the  Binomial  Theorem, 

(a  +  /3)"  =  a^  +  na^-^^  +  ^  ■^^-'^  a^^-^^  +  n-n-l-n-'l  ^^,3^3 

1  •  2  1  •  2  •  o 

where  w  is  a  positive  integer. 
The  binomial  coefficients 

n-n  —  l-n  —  2--'n  —  m-hl 
l'2-"m 
we  denote  by 

\mj 
We  have  obviously, 

'n 
n 

W\         /■       71      N  _  /'^  +  1 

m)      \m  —l)      \    m 
If  we  set  a  =  )S  =  1  in  1),  we  get 

2»=i+(«)+(«)+...+(^«i)+(:: 

If  we  set  a  =  1,  /S  =  —  1  in  1),  we  get 

n\   ,   fii\  .    ^      ^N«^W 

n 


0=1-  1  +  2  -•••  +  <:-i)"i 


It  is  often  convenient  to  set 
and 


\mj       ' 


if  m>n. 


Limits 

97.  We  extend  now  the  terms  sequence,  regular  sequence,  limit, 
etc.,  to  numbers  in  9?.  This  is  done  at  once  ;  for  the  definitions 
given  in  40,  42,  and  57  may  be  extended  to  0?,  by  simply  replacing 
the  term  rational  number  by  number  in  SR. 


62  IRRATIONAL   NUMBERS 

For  example,  the  sequence  of  numbers  in  9t 

«i,    «2j   «3>    •••  0 

is   regular  when,  for  each  positive  e  (not  necessarily  a  rational 
number  now)  there  exists  an  index  w,  such  that 

I  «„  —  a^  I  <  e, 

for  every  pair  of  indices  ?i,  v>m. 
Or  in  abbreviated  form,  when 

€  >  0,    m,    I  «^  —  a^  I  <  e.     n,  v>  m.  (2 

This  definition,  we  see,  is  perfectly  analogous  to  that  given  in 
57,  1  for  regular  rational  sequences.  Evidently  the  reasoning  of 
57,  3,  4,  can  be  applied  to  the  sequence  1).  Thus  the  e,m  test 
given  in  2  may  also  be  stated  in  the  form : 

e  >  0,    m,    I  «„  —  «„i  I  <  e,    w  >  m.  (3 

Similarly,  X  is  the  limit  of  the  sequence 

«!,    ao,    «3,    ••• 
when 

e  >  0,    ?7^,    I X.  —  «,J  <  e,    n  >  m.  (4 

As  before,  we  write 

X  =  lim  a„.    or    X  =  lim 


We  say  also  a„  converges  to  X  or  approaches  X  as  limit. 
This  may  be  indicated  by  the  notation 

«„  =  X. 

98.    Let  lim  a„  =  a  and  lim  ^^  =  /3. 

Then  lim  («„  ±  /3„)  =  «  ±  yS ;  (1 

lim  a„/3„  =  a/3.  (2 

//"yS,  ySj,  ySg,  •••=?^0,  ■^^^e  have  also 

lim-^  =  ^-  (3 

The  demonstration  is  precisely  similar  to  those  of  49,  50,  51 ; 
and  thus  does  not  need  to  be  repeated  here. 


LIMITS  63 

99.    We  prove  now  the  important  theorem  : 

Let  a  =  (aj,  a^,  •••),  the  as  ratiojial ;  then  lim  a„  =  a. 

We  must  show  that 

e>0,    wi,    |a  — a^]<e,    w>m.  ("1 

Since  the  sequence 

is  regular,  we  have 

cr  >  0,    m,    I  a„  —  a^  I  <  cr,    w,  1/  >  m.  (2 

Now  we  can  write,  by  60, 

Hence,  by  71,  supposing  n  to  be  fixed  for  the  moment. 
By  88, 

Hence,  by  2)  and  QQ,  2, 

I  a  —  a„  I  <  cr. 

Thus  if  we  take  o-  <  e,  we  have  1) . 

100.  If  a  sequence  . 

^  ^  =  «i,    «2'    ••• 

has  a  limit  X,  A  is  regular. 
For,  by  definition, 

e  >  0,    m,    I X  —  «,j  I  <  e/2,    n>m. 

\\  —  «^|<e/2,    v>m. 
Adding,  by  87,  3, 

I  '^re  "■  f^r  I  <  ^'         W,    y  >  7W. 

Hence  A  is  regular,  by  97. 

101.  1.    Conversely.,  if  ^  =  a^,  a^,  •••  is  a  regular  sequence,  there 
exists  one.,  avid  only  one,  number  «,  such  that 

lim  «„  =  a.  (1 

To  show  that  A  cannot  have  two  limits,  we  need  only  to  repeat 
the  reasoning  of  47, 


64  IRRATIONAL   NUMBERS 

We  show  now  A  has  a  limit. 
Let  8^,  §2'  ^3'  •••  (2 

be  a  sequence  of  positive  numbers  whose  limit  is  0.     We  choose 
the  S's  now,  such  that 

«,.  =  «»  +  ^«^     w=l,  2,  •••  (3 

are  rational.     This  is  evidently  possible  by  82.     The  sequence 

dfji    ^2'    ^3'     ***  C 

is  regular. 

For,  a^-a^=  a„  -«,+  (§„-  8  J .  (5 

Since  A  is  regular, 

e>0,  m,   |«„  — aj<-,    n,v>m.  (6 

A 

Since,  by  100,  the  sequence  2)  is  regular, 

\^n  —  ^v\<^l^'         n,v>m.  (7 

In  the  inequalities  6),  7),  we  may  take  m  the  same.     Then  5), 

Hence  4)  is  regular. 

We  set  a=  (a^,  a^^  •••). 

Then,  by  97, 

But,  by  3), 

Hence,  by  98, 


lim  a„  =  a. 


lim  a„  =  lim  a„  —  lim  B„ 


2.  As  a  result  of  1  and  100,  we  have  : 

In  order  that  a  sequence  a^,  a^^  •••  has  a  limits  it  is  necessary  and 
sufficient  that  it  is  regular. 


102.    Let  A  =  «!,  a^ 

sequence 


LIMITS  66 

be  a  sequence.     Let  us  pick  out  of  ^  a 

'r       '•2' 


where  t^  <  tg  <  ^3,  ••••     We  call  B  a  partial  sequence  of  A. 


EXAMPLES 


1. 


A  =  l, 

1 
2' 

1      1      1 

3'    4'    5' 

.., 

B  =  l, 

1 
3' 

1      1 

5'    7'  ■" 

C=l, 

1 

22' 

1     1 

23'    2*' 

•• 

D  =  l, 

_\_ 

1 

0  '      0      c'      < 

2-3      2.5     2.7 
Here  B,  C,  Z)  are  partial  sequences  of  A. 

2.  ^  =  1,    -,    1,    I,   1,    7,  — 

'2  3  4 

5=1,    1,    1,  ... 

^-2'    3'    4' 
5  and  C  are  partial  sequences  of  A. 

103.  1.  Among  the  symbols  given  in  42,  to  indicate  the  limit  of 
a  sequence 


.Ji.  CCj,     Cinj 


one  was 


lim  a„ 

A 


Analogously,  we  shall  denote  the  limit  of  a  partial  sequence 
B  z=  a,,   a^ .,   ... 


of  A,  by 


lim  a„. 

B 


2.  We  have  then,  obviously  : 

If  A  is  regular,  so  is  every  partial  sequence  B;  and 


lim  a„  =  lim  «„. 

A  B 


66  IRRATIONAL   NUMBERS 

3.  From  this,  we  conclude  at  once  : 

The  sequence  A  cayinot  be  regular,  if  it  contains  two  partial 
sequences  -S,  C,  such  that 

lim  «„  =^  lira  a„. 

B  c 

4.  The  sequence  A  cannot  be  regular,  if  it  contains  a  partial 
sequence  B  which  is  not  regular. 

5.  It  is  sometimes  a  difficult  raatter  to  show  that  a  sequence  A 
is  or  is  not  regular.  The  theorems  3,  4  enable  us  often  to  show 
with  ease  that  A  is  not  regular. 

Thus,  in  Ex.  2,  102, 

lim  «„  =  1,  lim  a„  =  0. 
B  c 

Hence  A  is  not  regular. 

6.  Unless  the  contrary  is  stated,  it  is  to  be  understood  that 

lim  «„ 

» 

has  reference  to  the  whole  sequence  A. 

104.  1.  From  98,  we  conclude  the  following  theorems,  which 
are  often  useful: 

If  lim  («„  ±  /3n)  =  o"i  and  lim  a„  =  a  ;  then  lim  /3^  exists  and  equals 
±a-T  a- 

2.  If  lim  a„/8„  =  tt,  and  lim  a^=:  a^O;  then  lim  ;S„  exists  and 
equals  7r/«. 

3.  If  lim  -—■  =  p,  and  lim  yS^^  =  /3 ;   then  lim  «„  exists  and  equals  ^p. 

Pn 

4.  If  lim  -^  =  p^O,  and  lim  a^=  a;   then  lim  y8„  ercis^s  and  equals 

The  demonstration  of  these  theorems  we  illustrate  by  proving  1. 

We  have  ^«  =  ±  (««  ±  /^n)  T  «„. 

Applying  98,  1), 

lim  ;S„  =  ±  lim  (a„  ±  yS„)  =F  lim  «„  =  ±  o"  T  «. 


LIMITS  67 

105.  1.  Let  \ixn  a^^  =  a  ;  let  ^,  ^  he  tivo  numbers^  such  that  ^<a<<y. 

Then  ^ 

P  <  (in  <  7-  t^  >  ni. 

The  demonstration  is  the  same  as  in  48. 

2.   Let  lim  «„  =  a,  and  «„  <  a.     Let  j3  be  any  number  <a. 

Then 

p  <  «„  <  a.  n  >  m. 

106.  1.  Let  lim  «„  =  « ;   if  )\.<a^<fi  for  n>m  ;  then 

X  <  «  <  /i. 
For,  suppose  a>  fx.     Let  /S  be  chosen  so  that 

/i  <  y8  <  «. 
Then,  by  105,  1,  a,,  >  yS.         n>m. 


Hence 

fin  >  /*» 

which  is  a  contradiction. 

2.  /f 

«„<x<A., 

awdf  ?[/ 

lim  «„  =  lim  yS„  =  /* 

then 

A.  =  yU. 

For,  by  1, 

yu.  <  X,    /u,  ^  \. 

Hence 

/A  =  \. 

107.    If 

«n<^«<7^i 

and 

lim  «„  =  lira  7„  =  \ 

then 

lim  8„  =  \. 

0 


For,  subtracting  a^  in  1),  we  get 

0</3„-«„<7„-«„.  (2 

Now  lim  (7„  —  «„)  =  lim  7„  —  lim  «^  =  \  —  X  =  0.  (3 

But  3)  states  that      e  >  0,  w,  7„  —  «„  <  e.         n>m. 
Then,  by  2),  /8„-«,<e. 


68  IRRATIONAL   NUMBERS 

This  relation  states  that 

lim(/3„-«J  =  0. 
As  lim  a„  =  X,  we  have,  by  104,  1, 

lim  ^„  =  X. 

108.  1.  A  sequence  J.  =  «p  a^,  •••,  whose  elements  satisfy  the 

relations  ^    ^ 

«« <  ««+i.    »*  =  1.  2, 

is  called  an  increasing  sequence. 

(^n>'\+i^   n  =  l,  2,  ... 

it  is  a  decreasing  sequence. 

If  it  is  either  one  or  the  other,  but  we  do  not  care  to  specify 
which,  we  may  call  it  a  univariant  sequence. 

If,  on  the  other  hand, 

««<««+^   -^  =  1,  2,  ... 
A  is  said  to  be  a  monotone  increasing  sequence. 

it  is  a  monotone  decreasing  sequence. 

If  A  is  either  one  or  the  other,  but  we  do  not  care  to  specify 
which,  we  may  call  A  a  monotone  sequence. 

Univariant  sequences  are  special  cases  of  monotone  sequences. 

2.  If  there  exists  a  fixed  positive  number  Gr^  such  that 
|a„|  <  (r,    w=  1,  2,  ••« 
A  is  said  to  be  limited^  otherwise  unlimited. 

109.  A  limited  monotone  sequence  is  regular. 

For  clearness,  let  A  =  ccj,   a^,  •  •  •    be   an  increasing  monotone 
sequence,  and  let 

a„<a.         n=l,2,-"  (1 

To  snow  that  A  is  regular,  we  must  show  that 

e>0,   w,    0<a„  — a^<e,         n>m.  (2 


LIMITS  69 


Since  A  is  monotone  increasing, 

0  <  «„  -  a^. 

To  show  the  rest  of  2),  take  tn^  at  pleasure.    Either  there  exists 
an  infinite  sequence  of  indices 

m^Km^Km^K-"  (3 

such  that  _ 

or  there  does  not. 

Suppose  such  a  sequence  3)  exists.     Then,  however  small  e  has 
been  taken,  we  can  take  the  integer  p  so  large  that 

Adding  the  first  p  inequalities  4),  we  get 
Hence,  by  5),  „     -^r 

"■nip  ^  "^1 

which  contradicts  1). 

We  thus  conclude  that  there  exist  but  a  finite  number  of  indices 
Wj,  such  that  4)  holds.     Thus  we  can  take  m  so  large  that 

««  —  ««i  <  e^         'W  >  ^h 
which  proves  tlie  other  half  of  2). 

110.   1.  A  limited  increasing  sequence  of  great  importance  is 

«,  =  (!+ 2)".         «  =  1,2,...  (1 

To  show  that  1)  is  increasing,  i.e.  that 

««  >  ««-i,  (2 

we  employ  the  relation  94,  5),  viz. : 

^»>a»-i[«-w(a-/S)].  (3 


Set 


a=l  +  -J_,    /3=1  +  -, 
n—\  n 


in  3)  ;  we  get  2)  at  once,  for  w  ^  2. 


70  IRRATIONAL  NUMBERS 


To  show  that  1)  is  limited,  we  set 

a=l  +  — -,   /3=1,        w  =  m4-l, 
zm 


in  3) ;  we  get 

l>ifl  +  ^T;        m  =  l,  2, 
2\       2mJ 

or  squaring, 

H^-£T- 

Thus 

«2«<4. 

But,  by  2), 

^2m-l<^2»i- 

As  all  positive 

integers  are  of  the  form 

2  m  or  2  m  — 1, 

4)  and  5)  give 

a„<4.         w  =  l,  2,  ••• 

(4 
(5 


2.  Since  the  sequence  1)  is  limited  and  monotone,  it  has  a  limit 
by  109.     We  set 

e=lim(l  +  -).         71=1,  2,3,  ••• 

As  the  reader  already  knows,  e  =  2.71828  •••,  and  is  the  base  of 
the  Napierian  system  of  logarithms. 

111.    1.   Let  A  =  a^,  Kg,  ■••  be  a  regular  sequence,  whose  limit  is  a. 
In  A  exist  partial  monotone  sequences  B  ;  and  for  each  such  sequence, 

lim  a„  =  a. 

B 

Then  are  two  cases:    1°  «  — «„,  n>m  has  one  sign,  when  not 
zero  ;    2°  a  —  «„  may  have  both  signs,  however  large  m  is  taken. 

Case  I.     To  fix  the  ideas,  suppose 

a— «„^0.         n>m.  (1 

In  this  relation,  it  may  happen  that  for  some  m'  >m 

a  — a„  =  0.         n^m'. 


LIMITS  71 

In  this  case, 

=  «,  a,  ... 

is  a  sequence  required  in  the  theorem. 

Let  us  suppose  now  that  there  are  in  1)  an  infinite  number  of 
indices  w„,  such  that 

«-«»,>  0. 

Let  Vj  be  one  of  the  indices  n^ ;   then 

Let  )8j  lie  between  these,  so  that 

a..</3i<a. 

Then,  by  105,  1,  and  103,  2,  there  are  an  infinity  of  elements  a„  , 
lying  between  ySj  and  a.     Let  a^  be  one  of  these,  so  that 

Let  /Sg  lie  between  a^  and  a ;  then  for  some  index  v^  we  have 

/Sg  <«.,<«. 
In  this  way  we  find  an  increasing  sequence 
B  =  a^.  «„,  «„,  — 


-1'     'J'     "«' 


which  is  a  partial  sequence. 

Then,  by  103,  2,  ,. 

lim  a^  =  a. 

n=oo 

The  number  of  sequences  B  of  this  type  is  obviously  unlimited. 

Oase  II.     Since  there  are  an  unlimited  number  of  indices  for 

which 

a  -  a„  >  0,  (2 

let  us  denote  those  values  of  n  for  which  2)  holds,  by  Wj,  Wj,  Wg,  •••. 
Then  the  partial  sequence  of  ^, 


72  IRRATIONAL   NUMBERS 

belongs  to  Case  I.     Hence  in  A'  lie  an  infinity  of  sequences  of  the 
type  B. 

2.  The  demonstration  of  1  shows : 

If,  in  the  regular  sequence  A  =  a^,  a^,  •••,  the  «„  do  not  finally 
become  all  equal,  there  exists  in  A  an  infinity  of  partial  univariant 
sequences  B  which  have  all  the  same  limit  as  A. 

The  Measurement  of  Rectilinear  Segments.     Distance 

112.  In  39,  43,  44,  we  have  made  use  of  the  graphical  representa- 
tion of  the  numbers  in  J?,  by  points  of  a  right  line.  We  wish  now 
to  extend  the  considerations  to  numbers  in  9?.  With  this  end  in 
view,  we  proceed  to  develop  the  theory  of  measurement  of  recti- 
linear segments  and  the  associated  notion  of  distance. 

113.  1.  Let  AB,  CD  be  two  rectilinear  segments.  We  say  ^5 
is  greater  than  CD,  when,  if  superimposed,  AB  contains  CD  as  a 
part ;  while  CD  is  said  to  be  less  than  AB.  If,  when  superimposed, 
AB  and  CD  coincide,  we  say  AB  and  CD  are  equal. 

2.  We  assume,  with  Archimedes,  that  if  the  segment  AB  is  laid 
off  a  sufficient  number  of  times  on  the  line  L,  we  can  obtain  a 

c  D        d"  t. 


segment  CD'  greater  than  any  given  segment  CD.    And  conversely, 
that  it  is  possible  to  divide  a  segment  CD  into  a  sufficient  number 


of  equal  parts,  so  that  one  of  them,  as  CE,  is  less  than  any  given 
segment  AB. 


MEASUREMENT  OF  RECTILINEAR  SEGMENTS.     DISTANCE       73 

114.  1.   Let  S  =  AB  be  a  segment  we  wish  to  measure  ;  and  let 
TJ  =  CD  be  a  segment  which  we  take  as  a  unit  of  comparison. 

If  we  ca^  divide  *S'  into  I  equal  segments,  equal  to  U,  i.e.  if 

we  say  I  is  the  measure  of  *S',  or  I  is  the  length  of  S. 

2.  If  it  is  impossible  to  do  this,  it  may  happen  that  n  segments 
S  are  equal  to  m  segments  U ;  i.e. 

n-  S=m'U. 
We  say  then,  that 

n 

is  the  measure  or  length  of  S. 

3.  In  both  cases  we  say  S  is  commensurable  with  U. 

The  segment  AB  being  commensurable,  we  say  its  length  I  ex- 
presses the  distance  of  A  from  B,  or  B  from  A.     We  write 

I  =  Dist  (A,  B), 

or  more  shortly  

l  =  AB. 

115.  We  show  now  that  the  number  I,  just  determined,  is  unique. 
This  is  evident  when  I  is  an  integer.     We  suppose,  therefore,  that 

nS  =  m  f/,  (1 

^"^  71^8  =m^U.  (2 

Multiplying  these  equations  respectively  by  Wj,  n,  we  get 
nn-^S  =  n^m  Z7,    Jin^S  =  nm^  U. 
.'.  n-^mU—  nm^U. 

.'.  n^m  —  nmp 

or  , 

m     m 

n      n' 
Thus,  the  two  equations  1),  2)  lead  to  the  same  value  of  I. 


74 

IRRATIONAL   NUMBERS 

116.    1. 

Let 

I 

oiS. 
nS 

s 

m 

.asure 

~n                                       ^ 

be  the  me 

Then 

=  mU.         (1          V 

Let  us 

divide 

U  into  n  equal  parts,  and  call  V 

one 

of  them. 

Then 

nV=U. 

This  in 

1)  gives 

nS  =  mn  V. 

Hence 

S=mV. 

This  shows  that  by  taking  a  new  unit  V,  whose  length  is  1/n  of 
the  old  unit,  the  length  of  S  can  be  expressed  as  an  integer. 

2.  The  above  considerations  also  give  us  a  new  way  for  defining 
the  length  of  S.  In  fact,  suppose  it  possible  to  divide  U  into  s 
equal  parts  V,  such  that 

S=rV.  (2 

Then 

Z=-.  (3 

s 

^°''  sV=U; 

hence,  multiplying  2)  by  s,  we  get 

sS=  rsV=  rU; 
so  that  the  length  I  oi  S  is  indeed  given  by  3). 

117.  Let  S  =  AB,  T  =  BC  be  two  segments  whose  lengths 
are  respectively  ^  „ 

b  a 

A B C 

a,  0,  c,  d  being  positive  integers 

If  we  put  them  end  to  end,  we  get  a  segment  W=  A  O  whose 

length,  we  show,  is  , 

n=^l-\-m. 

By  definition  we  have 

bS=aU,   dT=cU. 


MEASUREMENT  OF  RECTILINEAR  SEGMENTS.     DISTANCE       75 

Multiplying  these  equations  respectively  by  d,  6,  and  adding,  we 

sret 

^  hd'S+hd-T=adU+hcU=(iad  +  hc)U.  (1 

hd-S  +  hd-T=hd'W.  (2 

'Hence  1),  2)  give 

hdW=(ad  +  hc')U. 
Hence 

ad  +  hc      a  ,  c       ,  , 
"  =  -^-  =  4+5  =  '  +  ™- 

118.  1.  We  turn  now  to  the  measurement  of  segments  which 
are  incommensurable  with  the  segment  chosen  as  unit.  An 
example  of  such  segments  is  the  diagonal  of  a  square,  the  side 
being  taken  as  unit. 

I 1 


An    Bn 


To  measure  AB^  we  begin  by  marking  off  points  on  the  right 

line  L,  at  a  unit  distance  apart,  starting  with  A.     By  the  axiom 

of  Archimedes,  113,  2,  B  will  fall  between  two  consecutive  points 

of  this  set,  say  between  A^^  B^. 

Let 

Zj  =  Dist  (A,  A{). 

On  the  segment  A^B^  we  mark  off  points  at  the  distance  1/w 
apart,  where  n  is  an  arbitrary  positive  integer. 

Then  B  will  fall  between  two  of  these  points  which  are  con- 
secutive, say  between  A^,  B^. 

l^=  Dist  (^,  .42). 

We  may  continue  in  this  way,  subdividing  each  interval  A^,  B„i 
into  n  equal  parts,  without  end.  The  point  B  will  never  fall  on 
the  end  point  of  one  of  these  intervals,  for  then  AB  would  be 
commensurable.     The  sequence  of  rational  numbers 

^1'    ^2'      3'     *"  V-*- 

so  determined  is  monotone  increasing,  and  limited.     In  fact,  all 
its  elements  are  <  Zj  -f- 1.     Thus,  by  109,  the  sequence  1)  is  regu- 


k 


76  IRRATIONAL   NUMBERS 

lar,  and  so  defines  a  number  X.    We  say  X  is  the  measure  or  length 
of  AB,  and  we  write  as  before 


X  =  Dist  (A,  B)  =  A,  B. 

2.  If  we  had  taken  the  numbers 

IJ  =  I)ist (A,  B J,         /c  =  l,  2,  ... 

where  5^  denotes  the  right-hand  end  of  the  interval  in  which  B 
falls,  instead  of  the  numbers  l^,  we  would  have  got  a  monotone 
decreasing  limited  sequence 

whose  limit  X'  =  X. 

F«^'  l'-l=    1 

K 

whose  limit  is  0. 


n' 


K-V 


119.  We  have  defined  the  length  X  of  AB  by  a  process  which 
subdivides  each  interval  A^,  B^  into  7i  equal  parts.  The  question 
at  once  arises  :  would  this  process  lead  to  the  same  number  X,  if 
we  had  divided  each  interval  into  m  instead  of  n  equal  parts  ? 

We  prove  the  following  general  result :  Let  us  modify  the  above 
process  so  as  to  divide  the  first  interval  into  n^  equal  parts,  the 
second  interval  into  n^  equal  parts,  etc.  This  system  of  sub- 
division leads  to  a  sequence  which  we  denote  by 

The  limit  of  1)  being  X',  we  show  it  exists  and  X  =  X'. 

For,  each  point  A  J  will  fall  in  a  certain  interval  A  ,  B  of  the 
old  system  of  subdivision,  where  i^  is  the  lowest  index  for  which 
this  is  true. 

^^^^"^       Dist  (^^^  J  ^  Dist  (A,  AJ)  <  Dist  iAB^J. 

But,  by  118,  2, 

lim  Dist  (J.,  A^^  =  lim  Dist  (^,  B^J  =  X. 

Hence,  by  107,  ,.     ,^.  ,  .  ,     .   ,.      ^ 

•^  lim  Dist  (J.,  J.,„')  =  X. 


MEASUREMENT  OF  KECTILINEAR  SEGMENTS.     DISTANCE       77 

120.  The  process  explained  in  118,  119  is  obviously  applicable 
to  the  case  when  AB  is  commensurable.  The  only  difference  is 
that  after  a  certain  number  of  steps  the  point  B  may  fall  on  one 
of  the  end  points  of  the  little  segments  A„,,  B^.  In  this  case  the 
corresponding  sequence 

1'     ^2'      '"      ^S1     ^5'     ^si      '" 

would  have  all  its  elements  the  same  after  a  certain  one. 

121.  We  have  now  two  methods  for  measuring  a  commensurable 
segment;   viz.  those  given  in  114  and  120. 

Let  I  be  the  length  of  AB  as  given  by  114 ;  and  \  its  length 
according  to  120.     We  show 

l=\. 

Since  AB  is,  by  hypothesis,  commensurable,  we  have,  by  117, 
preserving  the  notation  already  employed, 

I  =  Dist  CAA,J  +  Dist  (A^B) 

<Dist(^A«)  +  Dist(^,A)-  0 

As  ^ 

Dist  (A„,B,n~) 


we  have,  from  1),  ^ 

where  Z,„  =  Dist  (J.,  A„^'). 

Passing  to  the  limit,  we  have,  since 

lim  ?,„  =  X,  lim  — —  =  0, 

n     ' 

l  =  \. 

122.    1.   Let  AC,  CB  be  any  two  segments  ;  we  show  that 

Dist  (AB^  =  Dist  (AC}  +  Dist  QCB). 
This  is  a  generalization  of  117. 


We  begin  by  supposing  that  one  of  the  segments,  AC,  is  com- 
mensurable, while  the  other,  CB,  is  not.       Then  AB  is  not  com 


78  IRRATIONAJ^   NUMBERS 

mensurable.  In  our  process  of  subdivision,  suppose  that  after  the 
mth  step,  B  falls  in  the  segment  B^,  BJ.  Then  AC  and  QB^  are 
commensurable.     Hence,  by  117, 

Dist  iAB^  =  Dist  (^  (7  )  +  Dist  (  05 J . 

In  the  limit,  we  get 

Dist  (^5)  =  Dist  (v4.(7)  +  Dist  ((75). 

2.  We  pass  now  to  the  general  case. 
After  the  mth  subdivision. 


Cm  Ci, 

let  Q  fall  between  G^  and   QJ^.     Then  AG^  is  commensurable. 
Hence,  by  1, 

Dist  (^5)  =  Dist  {A  (7,„)  +  Dist  (  G^B'). 

Passing  to  the  limit,  we  have 

Dist  (^5)  =  Dist  (J.  (7)  +  Dist  (  05) . 

Correspondence  hetioeen  9?  and  the  Points  of  a  Right  Line 

123.  1.  On  the  indefinite  right  line  L  mark  a  point  0  as  origin. 
Let  P  be  any  point  on  i,  and  let 

X=Dist((9,  P). 

If  P  lies  to  the  right  of  0,  we  associate  with  P  the  number  +  X  ; 
if  P  lies  to  the  left  of  0,  we  associate  with  it  —  X.  With  the 
origin  we  associate  the  number  0.  Thus  to  any  point  on  L 
corresponds  a  number  in  9^,  and  to  different  points  correspond 
different  numbers. 

2.  We  ask  now  conversely  :  does  there  exist  for  each  X  in  91, 

a  point  P  such  that  ,  _    _ 

^  |\l=Dist(0,  P)? 

For  all  rational  numbers  this  is  true  by  virtue  of  the  axiom  of 
Archimedes,  113,  2. 

Whether  it  is  true  for  every  number  in  9f,  cannot  be  demon- 
strated.    We  therefore  assume  with  Dedekind  and  Cantor  that 


CANTOR-DEDEKIND   AXIOM  79 

there  shall  exist  one  and  only  one  point  P  which  shall  lie  to  the 
right  of  (9,  if  X.  >  0,  and  to  the  left  of  0,  if  X  <  0,  and  such  that 

|X|  =  Dist(0,  P). 

This  we  shall  call  the  Cantor-Dedekind  axiom  or  the  axiom  of  con- 
tinuity of  the  right  line.  As  we  proceed,  it  will  be  made  evident 
that  many  apparently  simple  geometric  ideas  are  extremely  subtle 
and  complex.  One  of  the  most  elusive  of  these  is  the  notion  of 
continuity.  To  say  the  right  line  is  continuous  because  it  has  no 
breaks  or  gaps^  is  simply  to  replace  one  undefined  word  by  another. 

3.  We  have  now  established  a  one  to  one  correspondence  between 
the  numbers  of  '^  and  the  points  on  L.  We  may  consider  the 
points  as  images  or  representations  of  these  numbers. 

124.  1.  The  correspondence  which  we  have  just  defined  is  a 
generalization  of  that  given  in  35  for  R.  The  considerations  of 
39,  43,  44  can  now  be  extended  to  9^  without  any  further  com- 
ment. The  graphical  interpretation  of  sequences  and  their  limits 
which  we  thus  obtain  wall  illuminate  greatly  the  section  on  limits., 
97-111.  We  recommend  the  student  to  go  over  the  demonstra- 
tions which  we  gave  there,  employing  graphical  representations 
as  an  aid  to  the  reasoning.  Indeed,  some  of  the  theorems,  when 
interpreted  geometrically,  seem  almost  self-evident. 

2.   Consider,  for  example,  the  theorem  of  107. 

We  have  there  three  sequences,  A=  |a„|,  B=  |/S»|,  0—  {7»|. 

The  relation 

states  that  the  point  ^^  ^i^s  in  the  interval  /„=  («„,  7„). 

Since  now  both  end  points  of  i„  converge  to  the  point  \, 
evidently  any  point  in  /„,  as  /3„,  must  also  converge  to  the 
point  X. 

125.  As  another  example,  consider  the  theorem  of  109. 

To  fix  the  ideas,  let  J.=  ^«„|  be  a  monotone  increasing  sequence. 
The  points  in  the  figure  represent  a^,  ag'  "■'  ^®  have  drawn  a 
curve  through  them,  as  the  eye    seizes   more   easily  the   law  of 


80 


IRRATIONAL   NUMBERS 


increase  or  decrease  of  a  sequence  when  such  a  curve  is  drawn. 
The  reader  will  observe  that  since  the  sequence  is  monotone,  this 
curve  can  have  segments  parallel  to  the  axis  X.     As  A  is  limited, 

all  the  points  of  A  lie  be-     q 

tween  a  certain  line  G, 
and  a  line  JEJ  drawn 
through  the  first  point  a^ 
of  the  sequence.  To  see 
now  that  A  must  have  a 
limit,  let  us  suppose  the 
line  Gr  moved  parallel  to 
itself  toward  X.  Evi- 
dently there  is  a  line  F  below  which  G  cannot  move  without 
getting  below  points  of  A,  and  which  the  points  of  A  approach 
as  an  asymptote. 

If  A,  is  the  distance  of  F  from  X,  evidently 

lini  a„  =  X. 


Fig.  1. 


126.    As  a  final  example  of  the  helpfulness  of  graphical  repre- 
sentation, let  us  consider  the  theorem  of  111. 

The  two  cases  we 
considered  there  are 
represented  in  Figs.  1 
and  2.  The  heavy 
curves  represent  the 
law  of  increase  and 
decrease  of  the  sequence 
A.  The  points  «j,  a^, 
•'•  lie  on"  these  curves, 
but  are  not  indicated. 
The  straight  lines  A 
represent  the  limit  a  of 
A.  The  light  curve 
in  Fig.  1  indicates  an 
increasing  sequence  which  one  could  pick  out  of  A. 

By  the  aid  of  such  a  representation  the  theorem  becomes  almost 
self-evident. 


Fig.  2. 


CANTO R-DEDE KIND   AXIOM  81 

127.    1.  Let  A==\a,l  B  ={^J,  C  =  \y,\. 

Let  A  he  monotone  increasing,  and  C  moriotone  decreasing. 
Let 

f^n<^n<yn^  (1 

and  lira  (7„- «„)=0.  ■  (2 

Then  A,  B,  C  are  regular,  and  have  the  same  limit  X. 
Also 

G-raphically,  the  theorem  is  obviously  true. 
The  points  «„,  7„  determine  a  set  of  intervals 

-?»  =  (««^  7j. 

such  that  each  7„  lies  in  the  preceding  /„_i,  and  hence  in  all  pre- 
ceding intervals. 

By  2)  the  lengths  of  these  intervals  converge  to  0.  Geometri- 
cally, it  is  evident  that  the  end  points  a„,  7„  of  these  intervals  con- 
verge to  the  same  limiting  point  X,  and  that  any  sequence  of  points 
/3„,  where  /3„  is  any  point  in  /„,  must  also  converge  to  X. 

Arithmetically,  the  demonstration  is  as  follows : 

By  hypothesis, 

ai<«2<«3  "• 

7i>72>73-"  (4 

From  1), 

«n  <  7n- 

Hence,  by  4), 

«n<7r         w=l,  2,  ••• 

Thus  A  is  limited.     Similarly,  C  is  limited. 

Thus  A  and  C  are  regular,  by  109. 

Let 

lim  «„  =  X.  (5 

Then  2)  and  5)  give,  by  104,  1, 

lim  7„  =  X. 
Then,  by  107, 

lim  /3„  =  X. 


82  IRRATIONAL  NUMBERS 

2.  The  preceding  theorem  may  be  put  in  geometrical  language, 
and  gives : 

Let  Ij^  =  (a^^  7„)  he  a  sequence  of  intervals  w  =  l,  2,  3,  •••.  Let 
In  lie  in  -Z„_i,  and  let  the  lengths  of  these  intervals  converge  to  0.  Let 
/8„  he  any  point  iii  I^  (including  end  points').  Then  the  sequence 
f/3„|  is  regular,  and  all  such  sequences  have  the  same  limit  X.  The 
point  \  lies  in  every  I. 

3.  The  reader  should  avoid  the  following  error: 
Let  {a„|,  {/3„5  be  two  sequences,  such  that 

lim-(«„-^J=0.  (6 

Then  lim  «„,  lim  /3„  exist  and  are  equal. 

That  this  conclusion  is  false  is  shown  by  the  following  example  : 

«„  =  (-l)«  +  i,  ^„  =  (-l)^ 

Here  neither  limit  exists,  although  6)  is  satisfied. 

Dedekind's  Partitions 

128.  We  proceed  now  to  establish  the  notion  of  partition,* 
introduced  by  Dedekind,  to  found  his  theory  of  irrational  numbers. 

Let  a  be  any  number  of  9? ;  we  can  use  it  to  throw  all  numbers 
of  9"?  into  two  classes  A,  B.  In  A  we  put  all  numbers  <  a ;  in  ^ 
all  numbers  >a.  The  number  a  we  may  put  in  A  or  B.  This 
division  of  the  numbers  of  9?  into  two  classes  we  call  a  partition, 
and  say,  a  generates  a  partition  (A,  B).  Geometrically,  the  point 
a  may  be  used  to  throw  all  points  of  a  right  line  into  two  classes. 
In  class  A  we  put  all  points  to  the  left  of  « ;  in  ^  all  points  to 
the  right  of  a.      The  point  a  we  put  in  either  ^  or  5  at  pleasure. 

Example. 

Let  a  =  v^.     In  A  put  all  numbers  <  \/2  ;  in  B  put  the  numbers  ^  y/2. 
This  partition  may  also  be  generated  as  follows :  in  A  put  all  numbers  whose 
square  is  <  2  ;  in  B  all  numbers  whose  square  ^  2. 

*  The  German  term  is  Schnitt. 


DEDEKIND'S   PARTITIONS  83 

129.  1.  More  generally^  we  shall  say  that  any  separation  of  the 
numbers  of  9?  into  two  classes  J.,  B,  such  that 

1°  Any  number  of  J.  is  <  any  number  in  B, 
2°  Any  number  of  5  is  >  any  number  in  J., 
constitutes  a  partition  (^,  B^. 

2.  The  condition  2°  is  really  redundant,  as  it  follows  at  once 
from  1°. 

In  fact,  suppose  2°  did  not  follow  from  1° ;  i.e.  suppose  there 
were  a  number  /3  in  B.,  ^  some  number  a  in  A.  Then  there  is 
an  «  in  tI  which  is  not  <  any  number  in  J5,  for  «  is  not  <  /3. 
This  is  a  contradiction. 

3.  Two  partitions  (J.,  5)  and  (C,  i>)  are  the  same  or  equal 
only  when  A  and  C  contain  the  same  set  of  numbers;  or  only 
when  B  and  D  contain  the  same  numbers,  excepting  possibly  the 
end  numbers. 

130.  1.  We  consider  now  this  question :  suppose  a  law  given 
which  throws  all  numbers  into  two  classes  J.,  B^  such  that  every 
number  in  A  is  less  than  any  number  in  B,  and  every  number  in 
B  is  greater  than  any  number  in  A.  Is  there  a  number  X  in  9? 
which  will  generate  this  partition  (J.,  B^  ?     We  show  there  is. 

To  this  end  we  construct  two  sequences 

S=a-^.,    ttg,    "3,   ••• 

y  =  y8i,  yS^'  ySg,   -.• 

S  being  monotone  increasing,  and   T  monotone  decreasing,   as 
follows  : 

Let  ttj  be  any  number  at  pleasure  in  yl,  and  /3j  a  number  in  B. 

Their  arithmetic  mean  _ 

2 
lies  between  a^,  /3j. 

If  it  lies  in  A,  we  set 
if  it  lies  in  B,  we  set 


84  IRRATIONAL   NUMBERS 

We  build  now  the  arithmetic  mean  of  «2,  ^^.^  and  reason  with 
this  as  before.  Continuing  this  process  indefinitely,  we  get  the 
sequences  S  and  T. 

By  127,  2,  the  sequences  S,  T  have  a  common  limit,  which  we 
call  \. 

Let  X  generate  the  partition  (^',  B'). 

We  show  that  (^A,  B^  =  {A' ,  B-), 

by  showing  1°  that  every  number  in  A'  lies  in  A ;  and  2°  that 
every  number  of  A  lies  in  A' . 

1°.  Let  a'  4^\  be  any  number  of  A'.  By  105,  2  there  are  an 
infinity  of  numbers  «„  lying  between  «'  and  X.  But  a,^  is  in  A,  by 
hypothesis.      Hence  a'  <  «„  is  in  A. 

2°.    Let  «  be  any  number  of  A.     We  have  to  show  that  cc^X. 

Suppose  the  contrary,  X  <  «. 

Then  ^      ^ 

e  =  «  —  X  >  0. 

We  can  take  n  so  great  that 

A.-«.  =  ^^<-  (1 

On  the  other  hand,  by  supposition, 

«„  <  X  <  a  <  ^„. 
Hence 

which  contradicts  1). 

2.   We  have  thus  this  theorem  : 

Every  partition  can  he  generated  hy  a  number  in  $}?. 

131.  1.  A  partition  (^,  -S)  cannot  he  generated  hy  two  different 
numbers  X  and  yu,. 

To  fix  the  ideas,  let  X<  /jl. 

In  the  partition  (  C,  i>)  generated  b}^  /u,,  C  contains  all  numbers 
<  fi.  It  therefore  contains  numbers  >  X,  and  hence  numbers  not 
in  A.     Hence  {A,  J5),  ((7,  D)  are  different. 

2.  Since  each  number  generates  one  partition,  and  each  partition 
is  generated  by  one  number,  we  can  establish  a  uniform  or  one  to 


INFINITE  LIMITS  85 

one  correspondence  between  the  numbers  of  9?  and  the  aggregate 
of  all  possible  partitions. 

In  fact,  to  the  number  a  shall  correspond  the  partition  {A,  -6), 
that  a  generates.  To  the  partition  (i^,  (7)  shall  correspond  the 
number  X,  which  will  generate  {F,  6r). 

Infinite  Limits 

132.  Let  A  =  ttj,  ttg,  •••  be  an  unlimited  sequence  [108,  2].  The 
following  cases  may  occur  : 

1°.  For  each  positive  number  G,  arbitrarily  large,  there  exists  an 
m,  such  that  «„  >  G,  for  every  n  >  m.     In  symbols 

(r>0,     w,     a„>(x,     n>m. 

We  say  the  limit  of  A  is  plus  infinity/;  and  write 

lim  «„  =  +  oo  ,     lim  «,j  =  +  ao  ,     a„  =  +  go  . 

Such  sequences  are         i     o    o 

-L,  -^5  ^1  •" 

1!,  2!,  3!,  ... 

2°.  For  each  negative  number  (r,  arbitrarily  large,  there  exists 
an  m,  such  that  a„<  (r,  for  every  n^m.     In  symbols 

^ <  0,     w,     Uj^kG-,     ny^m. 

We  say  the  limit  of  A  is  minus  infinity  ;  and  write 

lima„=  — Qo,     lima„=  — QO,     a„ 

Such  a  sequence  is 

-10,     -100,     -1000,     ... 

In  both  these  cases,  we  say  the  limit  is  definitely  or  determinately 
infinite. 

3°.  The  elements  a„  do  not  finally  all  have  one  sign,  but  still 

lim  |«„|  =  4-  Qo. 

We  say  the  limit  of  A  is  indefinitely  or  indeterminately  infinite. 
Such  a  sequence  is 

1.     _2,     +3,     -4,     +5,     ... 


86  IRRATIONAL   NUMBERS 

133.  1.  Case  3  is  of  little  importance.  We  shall  therefore  in 
the  future,  when  using  the  terms  the  limit  is  infinite,  or  certain 
variables  become  infinite,  always  mean  definitely  infinite,  unless  the 
contrary  is  expressly  stated. 

The  symbol  +  qo  is  frequently  written  without  the  +  sign. 

The  symbol  ±  qo  means  that  the  limit  is  either  +  qo  or  -co, 
and  one  does  not  care  to  specify  which. 

The  limits  defined  in  the  previous  sections  are  called,  in  contra- 
distinction, finite  limits. 

The  symbols  +ao,  —  oo  are  not  numbers;  i.e.  they  do  not  lie 
in  ^.  They  are  introduced  to  express  shortly  certain  modes  of 
variation  which  occur  constantly  in  our  reasonings. 

2.  Finally,  we  wish  to  state,  once  for  all,  that  the  terms,  the  limit 
exists,  the  limit  is  X,  etc.,  or  an  equation  as 

lim  «„  =  \, 

always  refer  to  finite  limits,  unless  the  supplementary  phrase,  '■'■finite 
or  infinite,''^  is  inserted. 

134.  A  sequence  cannot  have  both  a  finite  and  an  infinite  limit. 
For,  if  ^  =  |a„|  has  a  finite  limit,  the  numbers  a„  lie  between 

two  fixed  numbers,  by  105,  1.     It  is  thus   limited.     It  cannot 
therefore  have  an  infinite  limit. 

135.  Let  A=  fa„},  and  let  B  be  any  partial  sequence  of  A.     If 

lim  «„  =  ±  00, 

A 

then  lima„  =  ±Qo. 

B 

The  demonstration  is  obvious. 

136.  If  the  limit  of  a  sequence  A=\a^\  is  indefinitely  infinite, 
its  positive  and  negative  terms  each  form  sequences  whose  limits  are 
respectively  +  go  and  —  qo  . 

For,  let  B=\^n\  be  the  sequence  formed  of  the  positive  terms 
of  A;  and  C=  |7„|  the  sequence  formed  of  the  negative  terms. 
By  hypothesis, 

(?>0,    m,    |a„|>(r,    n^m. 


INFINITE   LIMITS  87 

But  then,  for  a  sufficiently  large  m', 

Hence 

lim  ^n  =  +  ^1    lini  7„,  =  —  00. 

137.  Let  lim  «„  =  a,   lim  /3„  =  ±  oo  ; 
^A«w  1.                   lim(a„±/3„)  =  ±00,     lim^=0. 

3.  //«:^0,  ^>0,  aw(^  lim/3„  =  0; 

then  lim -^  =  4- 00. 

The  demonstration  is  obvious. 

138.  Let         «!,    a^,   ttg,    •••         ySj,    (^2'   ^3'    "• 

be  two  sequences. 

Let  I3n  ^cc„.         n  =  1,  2,  — 

If  lim  «„=  +CO, 
then  lim  /S„  =  +  oo . 

For,  by  hypothesis, 

G>0,   m,   an>  Gr,   n>m. 

Since  y3„  >  «„, 

we  have,  a  fortiori. 

Hence 

lim  ;8„  =  +  00. 

139.  Let  A  =  ttj,  ttgi  •••  ^^  <^  monotone  increasing  sequence.     Let 

he  a  partial  sequence  of  A. 

If  lim  «„  =  +  00,  then  lim  «„  =  +  oo, 

u  A 


88  IRRATIONAL   NUMBERS 

For,  by  hypothesis, 

Gr>0,    m,    a    >  G-.         n>m. 

're 

But  since  A  is  monotone  increasing, 

Hence  «r>  (r, 

and  lim  a^=  -\-  cc. 

140.   Let  Wj,  ^2,  ••'  Je  a  sequeyiee  of  integers  whose  limit  is  +go. 

Then 

lim  a™"  =  0,  zj  0  <  «  <  1.  (1 


For,  let  «>1. 
We  set 

Then,  by  94,  3), 

where 

We  apply  now  138. 
Since 

we  have,  from  4), 

Let  0<a<l. 
We  set 

Then 
Also 

As  by  3),  ,.       „,       ^ 

-'     ^  ■  Inn  p"'"  =  +00, 

we  have,  by  137,  , .       „,       ^ 

-^  lim  a'""  =  0, 

which  proves  1). 

The  truth  of  2)  is  obvious. 


=  1,  if  a=l.  (2 

=  +GO,   zy  a>l.  (3 

«  =  1  +  S.  8>0. 

«^  =  (i  +  sy«<>^^,  (4 

lim  /3^=  4-00, 
lim  a^  =  +(». 

1 

P>1. 

a™»  = 


INFINITE   LIMITS  89 

141.    We  consider  now  a  few  examples  involving  infinite  limits. 
Example  1. 

a^=l  +  l  +  l+..-+-.  ri  =  l,2,-. 

16  n 

Here 

lim«„=+Gfo.  (1 

For,  let 

/*=2™-l.         w  =  l,  2,  ... 

Then 


Om—l  ■•)m  1 

Each  parenthesis  is  >-• 

For, 

1,1     11      1 

-  +  ->-  +  -  =  -' 
2      3-^4     4      2 

l^l^l^l     1,1^1,1      1      , 

4  +  5  +  6  +  7>8-^8  +  8  +  r2'^*^- 
Thus 


^  m 

As 

lim  m=  +00, 

we  have,  by  138, 

lim  «   =  +  00 . 

But  \c(.^\  is  a  partial  sequence  of  the  increasing  sequence  la^]. 
Hence,  by  139,  we  have  1). 

142.   Example  2. 

«n  =  -  +  -T^  H —.  +  ••■  + 


a      a-(-l       a  +  2  a  +  w 

where  a=?^0,  —1,  —2,  —3,  ••• 

Then  ,.  ^ 

lira  «„=  +O0.  (J 

ie^  a>0. 

Then  there  is  a  positive  integer  m,  such  that 

w  —  1  <  a<m. 


90  IRRATIONAL  NUMBERS 

Then 

_J^>^.         ^7=0,  1,  2,  .^. 

Hence  ^  ^  ^ 

m     m+1  m+n 

But,  by  135,  141,  ,.      ^ 

Hence,  by  138,  ,. 

•^  lim«,j  =  +QO. 

Let  a<0. 

Then  there  exists  a  positive  integer  m,  such  that 

0<«  +  m<  1. 
Hence 

11  1,11 

a-\-m      a  +  m+1  (t-\-m-\-n  Z  n 

Then,  by  141  and  138, 

lim  7^=+ GO. 

But  {7„|  is  a  partial  sequence  of  Ja^}.    Hence  we  have  again  1), 
by  139. 


143.   Example  3. 


Q. 


_  «(«  + 1)  •••  («  +  »i) 


where  /S=9S=0,  -1,  -2,  ••• 

1°.    a>/3, /3>0.     Leta  =  a-/3. 

Then 

-— ! —  =  1  +  — m=0,  1,  •••w. 

p  +  71(1  p  +  m 

Hence 


by  90,  1.     Hence,  by  142, 

lim^„=+Qo. 
2°.    a>^,/3<0. 

If  a  is  a  negative  integer  or  0,  Q^  finally  becomes  and  remains  0. 


DIFFERENT   SYSTEMS   FOR  EXPRESSING   NUMBERS         91 
Hence  i-       /-»        n  n         ^  n 

Otherwise,  let  the  positive  integer  m  be  taken  so  large  that 

yS  +  m>0. 
Then 

Q  _«(«  +  l)"-rct  4-m  — 1)      («  +  m)  •••(«  +  n)  _  T>a 
^"     /3(/S+l)-..(/3  +  m-l)  *  (^  +  ^)...(;3  +  ri)~        "• 

The  first  factor  ^  is  a  constant.     In  iS'^,  set 

a'  =  a  +  m,  /3'  =  ^  +  m,  n—  m  =  n'. 
Then  ,^  ,       .^        .  ,         ,^ 

'^     yg'(^'  +  l)...(/3'+ri')* 

As  a'  >  13'  >  0,  iS'„  falls  under  case  1°. 

Hence  ..     ^ 

hm  ^„  =  ±  Qo, 

the  sign  being  that  of  B. 

3°.    a</3.     If  «=0,  -1,  -2,  ••.,  evidently  ^„  =  0. 

Let 

p  _ /3(/3  + l)'"(yS4- w)_  «  not  zero  or  a 

«(«  +  1)  ••  •  (a  +  n)  negati'/e  integer. 

Then  P„  falls  under  cases  1°  or  2°. 
But 

Hence  lim  ^„  =  0. 

4^    a  =  /3^0,  -1,  -2,  •••.     Here^„=l. 

Hence  ^■     ^       -, 

hm  ^„  =  1. 

Different  Systems  for  Expressing  Numhers 

144.    1.   Let  a  be  a  positive  integer,  and  m  any  integer  >1. 
Then  we  can  give  a  the  form 

a  =  a„m"  +  a!„_im"~^  H 1-  «o'  O 

where 

0<a«<m-l,         /«;  =  0,  1,  2,  .. 

and  w  is  a  positive  integer. 


92  IRRATIONAL   NUMBERS 

The  number  m  is  called  the  base  of  the  system.  The  base 
being  given,  the  numbers  a^,  «i,  •••  a^  completely  define  the 
number  a,  and  1)  may  be  written  more  shortly 

a  =  Cln^n-l  '"  %• 

When  m  =  10,  we  have  the  decimal  system.      For  example,  we 

^^'^*®  3 -104+1 '10^+7 -102 +  7.  10 +  9 

more  shortly  qiTTQ 

When  m  is  used  as  base,  the  numbers  a  are  said  to  be  expressed 
in  an  m-adic  system. 

2.  Let  0  <  «<  1.  With  a  is  associated  a  point  on  a  right  line  L, 
whose  distance  from  the  origin  of  reference  is  «.  To  measure  this 
distance,  let  us  divide  the  unit  interval  into  m  equal  parts,  each  of 
these  parts  into  m  equal  parts,  and  so  on.  Then,  as  shown  in 
118,  120,  57     7     7         ,  r9 


where 


a. 


1  —Hi 


m 
h=h  +  ^^  =  '^  +  %  (3 

^3=^2  +  ^3  =  ^  +  ^2  +  % 
m"*      m      m^     w^ 


The  numbers  Z^,  l^,  •■•  are  completely  determined  when  the  num- 
bers a  J,  ^2'  •'•  3^re  given.  Thus  «  is  determined  when  these  latter 
numbers  are  given,  and  instead  of  representing  a  by  the  system 
of  equations  2),  3),  we  may  employ  the  shorter  notation 

a  =  .a-^^a^a^  ••• 

For  example,  let  a  =  ^  and  m  =  10. 

Here  000 

a^  ^6,  ci^  =  o,  flfg  =  o,   ••• 

^""'^  i=.3333- 

which  is  the  familiar  decimal  representation  of  ^. 
If  we  take  w  =  5,  we  get  the  representation 

i=. 1313131- 


DIFFERENT   SYSTEMS   FOR  EXPRESSING   NUMBERS         93 

3.  Thus  every  positive  number  can  be  written  in  the  form 

where  the  as  and  5"s  are  ^  0  and  ^ m  —  1. 

4.  Certain  numbers  admit  a  double  representation,  in  an  m-adic 
system ;  viz.  those  numbers  in  which  the  6's,  after  a  certain  stage, 
all  equal  0  or  all  equal  m  —  1.     In  this  case  we  have 

a  =  a„a„_i  ■•■  Oq'  h^h^  ••■  J,  0000  •••  (4 

a  =  a„  ...  Uq  '  b^  ...  (bg—  l)(w—  1)(7W—  1)  •••  (5 

For  example,  when  m  =  1 0, 

23.5650000  ••• 

^^^  23.5649999 ... 

represent  the  same  number.  In  the  future  we  shall  suppose  that 
all  such  numbers  are  represented  by  the  form  4),  which  we  shall 
call  the  normal  form. 

5.  Numbers  of  the  type 

a„a„_i  •••  a^  •  b^b^  •••  b  000  ••• 

all  digits  after  b^  being  0,  are  usually  written  more  shortly,  by 
omitting  these  zeros.  Such  numbers  are  said  to  admit  a  finite 
representation. 

145.  The  expression  of  a  positive  number  iV  in  normal  form  in  an 
m-adic  si/stem  is  unique. 

1°.  Let  Nbe  an  integer.     We  show  first  that 

m^  >  «Q  +  a^m  +  ••  •  +  a„_im'*~^  (1 

This  is  obviously  true  for  w=  1.  We  apply  now  the  method 
of  complete  induction.  Supposing  1)  is  true  for  n  =  s,  we  show  it 
is  true  for  n=  s  +  1.     Let,  then, 

m^>aQ  +  a-^m  +  •••  +  a^^-^nf'^. 

Then,  since  both  numbers  on  the  left  and  right  are  integers, 

771*  —  (aQ  +  a^m  +  •  •  •  +  «5_iWi*~0  ^  !• 
Hence  ,+i     .      ,        ,  ,_ix    ^ 


94  IRRATIONAL   NUMBERS 

Subtracting  an  integer  5=0,  1,  •••  m  —  1,  from  both  sides, 

w'+^  —  (5  +  a^m  +  a^m"^  + h  as_inf)  >  0. 

Hence,  changing  the  notation  slightly, 

w'+^  >  «o'  +  «i'^M h  ajm%  (2 

if  the  a'  are  ^m  — 1. 
This  established,  let 

iV=  5(j  +  5jm  +  Sgm^  +  •  •  •  +  b^m^ 

where  a^^^O,  b^^O. 

Then  1)  shows  that  if  r>s,  then  M^JST. 

Hence,  if  Mia  to  equal  iV",  it  is  necessary  that  r  =  s. 

If  r  =  s,  then  1)  shows  that  iltf^iV",  according  as  a^^bj..  Thus, 
if  M=  N,  it  is  necessary  that  a,.  —  b,..  In  this  way  we  may  con- 
tinue, and  so  show  that  when  M=  N, 

a^  =  b^,     a;=  0,  1,  ...  r. 

2°.  Let  Q<N<1. 
Suppose  T^T 

^^  N=  .aja2""^r-l^r"* 

jT    — •   •  lA/'t  (Xn  •  "  *  Ct«. Y^f  •  •  * 

where  a^^  5^.     To  fix  the  ideas,  let  a^  >  5^.     Then  iV>/*. 
For, 

Since  P  is  written  in  the  normal  form,  there  exists  an  s^r, 

such  that  ,  ^ 

bg<m  —  \. 

Then  ,  ,       . 


But  since  a^.  >  5^, 


2  -^-^  2- 


Hence  .    iVj  +  iVg  >  iV^j  +  Pg' 

and  a  fortiori^  ^     p 


CHAPTER   III 
EXPONENTIALS   AND  LOGARITHMS 

Rational  Exioonents 

146.  Having  developed  now  the  number  system  9?  with  suffi- 
cient detail,  we  shall  in  this  and  the  subsequent  chapters  represent 
numbers  in  9t  indifferently  by  Greek  and  Latin  letters. 

147.  Up  to  the  present  we  have  defined  the  symbol 

a*"  (1 

only  for  positive  integral  values  of  the  exponent  fi.  We  proceed 
to  define  it  for  any  value  of  /a,  supposing  a  >  0.  We  begin  with 
rational  values.    The  numbers  1)  are  then  called  roots  or  radicals. 

148.  1.  Let  a>0,  and  let  n  he  a  positive  integer.  There  exists 
one  and  only  one  positive  number  satisfying 

x^  =  a.  (1 

Let  (^B,  C)  be  a  partition  such  that  B  contains  all  positive 
numbers  h  such  that  b"'  <  a. 

Let  p  be  the  number  which  generates  (.6,  (7),  130,  2.  By  130, 
1,  we  can  pick  out  of  5  a  monotone  increasing  sequence  \b^\  and 
out  of  C  a  monotone  decreasing  sequence  {c^}  such  that 

lim  b^  =  lim  c^  =  p. 

As  l>l<a<  C 

we  have,  by  106,  2, 

lim  6^,  =  a. 

95 


96  EXPONENTIALS   AND  LOGARITHMS 

we  have 

p"  =  a.  (2 

Hence  p  satisfies  1). 
There  is  but  one  positive  solution  of  1).     For  if 

rr'^  =  a.,  (3 

we  have,  from  2),  3), 

p""  =  a\ 
Hence,  by  75,  3, 

p  =  (T. 

2.  We.  write  i 

p  =  ^a  =  a^- 

3.  When  n  is  odd,  1)  has  only  one  solution  in  9^,  viz.,  x  =Va,- 
When  n  is  even,  it  has  two  and  only  two  solutions  in  9?,  viz., 

Va,  —  V  a. 

149.  1.   From  the  preceding  we  have  readily: 

Let  a  <  0.      Then 

x^  =  a 

has  no  solution  if  n  is  even,  and  if  7i  is  odd,  it  has  one  and  only  one 
solution,  viz.,  —-yj—a. 

For  brevity  we  often  write,  when  n  is  odd  and  a  <  0, 

V«     tor     —  V—  «. 

When,  however,  a  >  0  the  radical  -y/a  shall  always  be  a  positive 
number. 

2.    The  equation  x^  =  0   admits   one  and  only  one  solution,  viz., 
X  =  0,  in  $R.      We  write  i 

70  =  0^=0. 

150.  1.  If  m,  n  are  positive  integers,  and  a>0,  then 

1  1 

.      (a")™  =  (a"'y.  (1 

Set  1 

p  =  a'^K 


RATIONAL   EXPONENTS  97 

Then 

p«  =  a, 

and  1 

r  =  («")"•;  (2 

also 

The  equation  3)  shows  that  p"^  is  the  positive  solution  of 

a;"  =  a*". 
Hence,  by  definition,  i 

p'«  =  {dPy.  (4 

Comparing  2)  and  4),  we  have  1). 

2.   We  write  i  i        m 

{ccy  =  (a"')"  =  a". 

We  have  now  the  definition  of 

a*",         a  >  0 
for  positive  rational  exponents. 

151.  Let  ft  be  a  positive  rational  number.     We  define  the  symbol 
a"**  by  the  relation  ^ 

a-'^  =  — . 

We  also  set  „      ^ 

a"  =  1. 

152.  ie^  r,  s  5e  rational  numbers,  and  a  >  0.      ?%gW' 

a'-a*  =  a'-^'.  (1 

This  equation  expresses  the  addition  theorem  for  rational  ex- 
ponents.    It  is  a  generalization  of  74,  2. 
To  fix  the  ideas,  suppose  r,  s  >  0.     Let 

I  m 

r  =  — ,      s  =  — , 
n  n 

where  I,  m,  n  are  positive  integers. 

Let 

p=a'^  ;    then  p""  =  a'.  (2 

r  of 

a  =  a'  ;    tjien  o-"  =  a'".  (3 


98  EXPONENTIALS  AND   LOGARITHMS 

Multiplying  2),  3),  we  get 

{pa-y  =  «'+'". 
This  shows  that  />  •  o-  is  the  positive  root  of 

Hence  i+m 

p(T  =  a  '^  .  (4 

But  2),  3)  also  give 

pa  =  a^'a\  (_5 

Comparing  4),  5),  we  get  1),  since 

Z  +  m 

=  r  +  s. 

n 

153.  Let  fi  be  a  rational  number,  and  a  >  0. 

Then  „      ^ 

a'">0. 

171 

For  let  ytt  =  —  >  0 ;  m,  n  positive  integers. 

We  have,  by  148,  i 

a">0. 

Hence  ^       l      l        I 

(t  =z  a^  '  a''  ••'  a" >  0.     m  factors. 

If  fjL<Q,  let  fi  =  —  v,v>0. 

Then  i 

a'"  =  — , 
a" 
and  as  a"  >  0,  so  is  a*^  >  0. 

If  /u,  =  0,  we  have  a'^  =  1,  by  151. 

154.  Let  fjL  be  a  positive  fraction  and  a>0.      Then 

a'^  =  1  according  as  a  =  l. 

Let  /i  =  — ;  m,  n  positive  integers. 

1 
If  a  >  1,  then  a"  >  1. 
For,  suppose  the  contrary,  i.e.  let 


RATIONAL   EXPONENTS  99 

Raising  to  the  nth.  powers,  by  75,  2, 

a<l, 
which  is  a  contradiction.     Hence 

Ifa>l,  a">l. 

m 

Hence  a"  >  1,  by  75,  2. 

The  other  cases  are  similarly  treated. 

155.  Let  n  be  a  positive  integer  and  CL>0.      Then  a^^ a  according 
as  a  =  l. 

Let  a  <  1  and  suppose  2 

Then,  by  75,  2, 

a^a\  (1 

But  when  a<l, 

a"<a, 

by  75,  4.     This  contradicts  1). 
i_ 

Thus,  when  a <  1,  a"  >  a. 

The  other  cases  are  easily  treated  now. 

156.  1.  Let  «>0  and  let  /m  be  a  positive  fraction.     If 

a>'>b>0, 

then  I 

a>b''.  (1 

For,  let  /A  =  — ;  w,  w  positive  integers. 
n 

From  TO 

a"  >b 

follows,  by  75,  2, 

a'">J™.  (2 

Suppose  now  1)  were  not  true,  i.e.  suppose 

.      Then  _ 

a'^  <  J", 
which  contradicts  2). 


100  EXPONENTIALS  AND  LOGARITHMS 

2.   Let  a  >  0  and  let  fi  be  a  negative  fractioyi. 
If  a^>b>  0, 

then  a<.b^.  (3 

We  can  set  fi  =  —  v^  v  >  0 
Then  -j 

a- 
This  reduces  2  to  1. 

157.  Let  fjL<p  be  two  rational  numbers  ;  let  a  >  0, 

Then  .  7.  \-,  ^-1 

a^^a^  according  as  a^l.  (^1 

We  can  set 

r  s 

u  =  -     V  =  -t 

t  t 

where  r,  s,  ^  are  integers  and  ?  >  0. 

To  fix  the  ideas,  let  a  >  1,  and  /a,  i'>  0. 
Suppose  for  this  case,  1)  were  not  true,  i.e.  that 

a*^  ^  a". 
Then 

a''  ^  a^  by  75,  2, 
which  is  absurd,  since  r<s. 

Thus  1)  is  true  for  this  case.     In  the  same  way  we  may  treat  the 
other  case. 

158.  Let  jxbe  a  rational  number .,  and  let  a„  >  0,  w  =  1,  2,  •••. 

If  lim  a„  =  1,  (1 

then  lim  <  =  1.  (2 

To  fix  the  ideas,  let 

r  ... 

/i,  =  -  ;  r.,  s  positive  integers. 

If  «„<1,     «„<«,/<!; 

1 

if  a„>l,      1  <«„"<«„, 

1 
by  154,  155.     Thus  in  either  case  a,j*  lies  between  1  and  a„.     Ap- 
plying 107,  we  have,  using  1), 

lim  aj—  1. 


IRRATIONAL   EXPONENTS  101 

Since  r         i     i         . 

«/=«„*■«,/•••«„%         r  factors. 

we  have,  by  98, 

?• 

lim  a„^  =  lim  <  =  1. 
The  other  cases  are  now  easily  treated. 

159.  Let  n  he  a  rational  number;  let 

lim  a^  =  a  >  0,         a„  >  0. 

Then 

hma'^=  ai^.  (1 

For, 

<  =  «.(^J.  (2 

But,  by  hypothesis, 

lim^=l,  by  98. 

a 

If  we  apply  now  158  to  2),  we  get  1). 

Irrational  Exponents 

160.  Let  R  =  r J,  r^^  •-•  he  a  sequence  of  rational  numbers  whose 
limit  is  0.      Then^  if  a  >  0, 

lima''»  =  l.  (1 

We  show :  /^  i  -i        .  i  ^o 

e  >  0,    7n,    1 1  —  a'^"  I  <  e,    w  >  Tn,  (2 

which  is  the  same  as  1). 

Let  a  >  1,  r„  >  0  ;   then,  by  154, 

a'^">l. 

Since  r„  >  0  and  as  small  as  we  please,  n  being  sufficiently  large, 
we  can  take  m  so  large  that 

1 

—  >g,        n>m, 

rn 

however  large  the  positive  integer  g  be  chosen. 
But,  by  91,  3),  ^^       ^       ^ 


102  EXPONENTIALS   AND   LOGARITHMS 

We  can  also  take  g  so  large  that 

1  +g€>a. 

Then  ,-.       x„ 

(l  +  ey>a.  (3 

On  the  other  hand,  by  157, 

(l  +  e)^«>(l  +  6)^.  (4 

Hence  3)  and  4)  give 

(l-fe)'"">a.         n>m. 

This  gives,  by  156,  1), 

1  +  e  >  ar\ 

Hence  2)  holds  in  this  case. 

Let  a  <  1,  r„  >  0,     We  set 

1 

a=-; 

0 

then 

b>l. 

By  the  preceding  case 

6''«<l  +  e. 

Hence 

a^''=^>-^>l-e, 

J'^"      1  +  e 

by  89,  1).     This  gives 

1  —  a^«  <  e. 

Hence  2)  holds  in  this  case. 

Let  r„  <  0.  This  case  reduces  to  the  case  that  r„  >  0,  by  observ- 
ing that 

We  consider  now  the  case  that  the  r^  do  not  all  have  one  sign. 
We  divide  li  into  three  sequences,  Mq,  M+,  R_.  In  the  first,  we 
throw  all  r„  =  0 ;  in  the  second  all  r„  >  0  ;  in  the  third  all  r„  <  0. 
Should  any  of  these  sequences  contain  only  a  finite  number  of 
elements,  it  can  be  neglected.  For  we  have  only  to  consider  a 
partial  sequence  of  R,  obtained  by  omitting  the  terms  of  R  up  to 
a  certain  one. 


IRRATIONAL   EXPONENTS  103 

Consider  ^+.  We  have  seen  there  exists  in  it  an  index  m'  such 
that  2)  holds  for  every  n>m' . 

Similarl}^  in  jB_,  there  exists  an  index  m"  such  that  2)  holds 
for  every  n>m" . 

Consider  finally  R^.     As  r„  =  0,  2)  holds  for  every  n  of  R^^. 

Thus  if  m  be  taken  >w',  m" ,  2)  holds  for  every  n>m'va.  R. 

161.  Let  A  —  aj,  a^^  •••  he  a  regular  sequence  of  rational  numbers, 
and  let  5  >  0.      Then 

b%  5%  •••  (1 

is  regular. 

We  have  to  show : 

e>0,  m,  15""— 5'''»|<e,        n>m.  (2 

Set 

d^  =  6"»  -  b"'"  =  5«"'(6««-«'»  -  1).  (3 

Since  A  is  regular,  we  have 

8>0,  m,  |a„— a,„|<8.       n>m. 

But  if  8  is  taken  small  enough,  by  160, 

|5a«-«m_l|<^,  (4 

where  ?;  >  0  is  arbitrarily  small. 

Since  A  is  regular,  there  exist,  by  65,  5,  two  rational  numbers, 

Q,  R,  such  that 

Q<a^<R.         n=l,  2,  ...  (5 

Then  3)  gives,  by  4)  and  5), 

\dn\<b^r],  if  b>l; 

<b%  if  5<1,  by  157. 


(6 


If  6  >  1,  we  take  i]  = 


€ 


5^' 

if  5  <  1,  we  take  '^  —  To 

Then  in  either  case  2)  holds. 

The  case  that  5  =  1  requires  no  demonstration. 


104  EXPONENTIALS   AND   LOGARITHMS 

162.  Let  ap   a^^  ■•■   and  a^,  a.^,  ••    he  two  sequences  of  rational 
numbers  having  the  same  limit.      Then,  if  5  >  0, 

lim  b""  ~  lim  5"».  (1 

By  161,  both  limits  in  1)  exist. 

Let  c?^  =  6"«  -  6"«  =  5«"(1  -  J""-"").  (2 

We  have  only  to  show  that 

lim  d^  =  0.  (3 

But 

lim  (a„  -  «„)  =  0. 
Hence,  by  160, 

lim  (1  -  5»»-«'')  =  0.  (4 

Hence,  2),  4)  give  3). 

163.  We  are  now  in  the  position  to  define  irrational  exponents. 
Let  ,  . 

/I  =  (7-1,  7-2,   •••) 

be  a  representation  of  fi.     We  say 

a*"  =  lim  a''".  (1 

By  161,  the  limit  on  the  right  of  1)  exists ;  by  162,  it  is  the 
same  whatever  representation  of  /x  is  taken. 

164.  1.  Let  r^,  r^,  •••  be  a  sequence  of  rational  numbers  having  a 
rational  limit  r.      Then,  if  5  >  0, 

lim  b''^  =  b\  (1 

In  fact,  the  sequence 

r'l,  r'a,  r'3,    •••;         'r'^  =  r,         w=l,  2,  •• 

has  r  as  limit. 

^         '  hm  5%  =  hm  h'n 


But 

lim  h^n  =  lim  b^  =  b'^. 


(2 


This  in  2)  gives  1). 

2.  The  object  of  1  is  to  show  that  the  definition  of  a*^  given  in 
163  does  not  conflict  with  that  given  in  150,  151,  in  case  /m  is 
rational. 


IRRATIONAL  EXPONENTS  105 

165.   1.    Let  fi  be  an  arbitrary  number,   and  r,  s,  two  rational 
numbers,  such  that  r  <  /u,  <  s.      Then  for  5  >  0, 

b^'^b^^  b%  according  as  b^l.  (1 

For,  let 

fi  =  (mj,  m^,  •••), 
the  m,  s  being  rational. 

Then,  by  105,  1, 

r  <  m„  <s.         n>v. 
Hence,  by  157, 

b'-<b"'n<b%        if       b>l. 

Then  passing  to  the  limit,  by  106,  1, 


b'<b>'<  b\ 

(2 

Here 
another 

the  equality  sign   must   be   suppressed, 
rational  number  such  that 

For,  let  r'  be 

Then, 
But 

r<r'<fji. 
as  in  2), 

b"-  <  b''. 

b'  <  b'\ 

(3 

by  157. 

From  3),  4)  we  have 

b'-  <  b'^. 

Thus  the  equality  sign  in  2)  must  be  suppressed. 

The  truth  of  1),  when  6<1,  follows  in  a  similar  manner. 

2.   As  a  corollary  of  1,  we  have : 

Let  a  >  0  ;  then  a*^  vanishes  for  no  value  of  fi. 

In  fact,  the  relation  1  shows  that  a'^  always  lies  between  two 
positive  numbers,  by  153. 

166.  1.  The  properties  given  in  the  preceding  articles  for 
rational  exponents  hold  for  irrational  exponents  also.  We  illus- 
trate the  demonstration  in  a  few  cases. 

Let  X  <  yu.,  and  5  >  0.      Then 

5'^ ^6%  according  as  b^l.  (1 


106  EXPONENTIALS  AND   LOGARITHMS 

To  fix  the  ideas,  suppose  6  >  1.     Let  r  be  a  rational  number, 

such  that 

\<r<iJb. 

Then,  by  165, 
Hence 

2.   As  corollary  we  have  : 
If  6  >  1,  we  conclude  from 


that 


whereas,  if  0<b<l,  we  conclude  that 
3.  In  1)  let  X  =  0.     Since 


^  < 


b^=l 


we  have  r 
Let 


/i  >  0,  and  6  >  0  ;  then 
b'^  =  1,  according  as  6  =  1. 

167.  *""=f-         *>0- 

For,  let 

Then 


—  «  =  (— ffj,  —  ^2,  "Oi  by  71,  3. 


Since,  by  161, 

we  have 

6-»=  lim6-''™  = 


lim  5""      6*' 
since  5*  :^  0,  by  165,  2. 


IRRATIONAL   EXPONENTS  107 

168.   If  a>0. 

This  is  the  addition  theorem  for  any  exponents,  and  is  a  generali- 
zation of  152. 


Let 

Then 

Hence 


\  =  lim  X„,  ^l  =  lim  /a„. 
a^  =  lira  a-^",  a*^  =  lim  a*^". 
a^a'^  =  lim  a^^^  lim  a*^"  =  lim  a^"**^" 
=  lim  a^n+f*",  by  152, 


169.    iei  Xj,  Xg,  •••  5g  a  sequence  whose  limit  is  +  go, 

lfa>0, 

+  CO,  ?/  a>l, 


lim  a^n=  -, 

0,  ?/  «  <  1. 

For,  let  a>l. 

Let  ^„  be  the  greatest  integer  in  X„. 

Since  lim  X„  =  +  oo,         lim  ^„  =  +  oo. 

Then,  by  140, 


lim  a'n  =  +  00. 

As 

lim  a'^'i  =  +  00, 

by  138. 

Let  a< 

1. 

Set 

h. 

=  -.         Then  6 

The  demonstration  follows  now  at  once. 

170.    Let  aj,  a^,  •••  he  a  sequence  of  positive  numbers  whose  limit 

is  1. 

Then 

lim  a„^  =  1. 


108  EXPONENTIALS   AND   LOGARITHMS 

Let  r,  s  be  rational  numbers,  such  that 

r<\<s. 
Then,  by  166,  1, 

a/  <  «/  <  <^       «n  >  1 ;  O 

an  <  a/  <  <^         «n  <  1-  (2 

Let  us  apply  now  107.     Since,  by  158, 

lim  ttjf  =  lim  a,/  =  1, 

we  have,  from  1),  2), 

lim  a/  =  1. 

171.  Let  aj,  a^,  •••  be  a  sequence  of  positive  numbers  tvhose  limit 
is  a  >  0.      Then 

lim  a^^  =  «^.  (1 

Since,  by  170, 

lim^=l, 
a 

1)  follows  from  2)  at  once. 

172.  Let  \j,  Xj'  •■*  ^^  ^  sequence  whose  limit  is  X.     7/  a  >  0, 

lim  a^«  =  a^.  (1 

For,  let 

^11     ^2'     '"•>  ^1'     ^2'      °" 

be  two  sequences  of  rational  numbers  whose  limits  are  X,  and  such 
that 

'>'n<K<Sn^  71=1,   2,   ••• 

To  fix  the  ideas,  let  a  >  1 ;  then,  by  166, 

a^'n  <  a'^n  <  a'n.  (2 

By  162, 

lim  al'n  z=  lim  a^n. 

The  application  of  107  to  2)  gives  1). 
The  case  that  a  ^  1  is  now  readily  treated. 


LOGARITHMS  109 

Logarithms 

173.    Let  a,  5  >  0,  and  b^l.      The  equation 

h'  =  a  (1 

has  one,  and  only  one,  solution. 

To  fix  the  ideas,  let  6  >  1.     We  form  a  partition  ((7,  i))  in  which 
C  contains  all  numbers  c,  such  that 

h'<a; 

while  D  contains  all  numbers  d,  such  that 

h'^>a. 

This  separation  of  the  numbers  of  9?  into  the  classes  C,  D  is 
indeed  a  partition.     For  every  number  of  C  is  <  any  number  of  B. 
In  fact,  from  ^^  ^  ^,^ 

follows,  by  166,  2, 

Let  ^  be  the  number  which  generates  (C,  i)) ;  let 


^l>^2 


>... 


be  the  monotone  sequences  of  130,  whose  common  limit  is  ^. 

Tlien  ,, 

o«  =  a.  (2 

For,  by  171,  t     7.       i-     7^       7t  xo 

On  the  other  hand,  ,  ,v 

b'n<:a<  ¥n.  (4 


From  3),  4)  we  have,  by  106,  2, 

lim  ¥n  =  a.  (6 

From  3),  5)  we  have  2). 

The  equation  2)  shows  that  |  is  a  solution  1).      Let  r)  be  also  a 

solution,  so  that 

5''  =  a.  (6 

From  2),  6)  we  have 

b^  =  b\ 

Hence,  from  166,  2, 


110  EXPONENTIALS    AND   LOGARITHMS 

174.  1.   As  we  have  just  seen,  the  equation 

1^^=  a,         a>0;         6>0  and  =#=  1, 

admits  one,  and  only  one,  solution.      This  uniquely  determined 
number  |,  we  call  the  logarithm  of  a,  the  base  being  b;  and  write 

I  =  logj  a, 

or  when  we  do  not  care  to  indicate  the  base, 

I  =  log  a. 

2.  We  shall  suppose,  once  for  all,  that  the  base  b  is  ^  1 ;  also 
that  the  numbers  whose  logarithms  we  are  considering  are  >  0. 

3.  From 

log  u  =  log  w, 
follows 

u  =  v. 

The  demonstration  is  obvious. 

175.  log  ab  =  log  a  +  log  b. 

This  is  the  addition  theorem  of  logarithms. 
Let  the  base  be  c.     If 

(t  =  log  a,    ^—  log  b, 
then 

c«  =  a,    c^  =  b. 
Multiplying,  we  have 

c'^c^  =  c'^+P  =  ab. 
From  the  equation 

c"+^  =  ab, 
we  have 

log  a6  =  a  +  /3  =  log  a  +  log  b. 

176.  By  using  the  properties  of  exponentials  we  may  deduce  in 
a  similar  manner  all  the  ordinary  properties  of  logarithms.  As 
this  presents  nothing  of  interest,  we  pass  on.  We  note,  however, 
the  following  important  relation. 

Let  a  >  0,  and  b  be  the  base  of  our  logarithms.      Then 


LOGARITHMS  111 

For,  by  definition,  jioga^^^^^  (2 

But  .  , 

log  a>^  =  II  log  a. 

This  in  2)  gives  1). 

177.    Let  aj,  a^,  •"  he  a  sequence  whose  limit  is  1.      Then 

lim  log  a„  —  0.  (1 

To  fix  the  ideas,  let  5,  the  base  of  our  logarithms,  be  >1. 
Let€>0,  then,  by  166,  3, 

b^>l.  (2 

Hence 

g=l-->0.  (3 

Since  , .  ^ 

hm  a„  =  1, 

we  have  _      ^  _  ^      c. 

6 >  0,    m,    —  6 <  a„  —  1  <  0,    n>m; 

which  gives  -,       c^  -.       c^  ^a 

^  1  -  S  <  a„  <  1  +  S.  (4 

From  3)  we  have  ., 

1-8  =  1. 

b' 


This  in  4)  gives  ^ 

On  the  other  hand, 


«-,>-  (5 


by  3),  2). 

This  gives  l  +  8<6..  (6 

Then  6)  and  4)  give  ^_^^j._  ^^ 

From  5),  7)  we  have  finally 

This  may  be  written,  by  176,  1), 

b-'<¥°^"''<b^. 
Hence,  by  166,  2, 

—  e  <  log  a„  <  e,         n>m, 

which  is  another  form  of  1). 


112  EXPONENTIALS   AND   LOGARITHMS 

178.    Let  Vun  a^  =  tt>0.      Then 

lirn  log  a„  =  log  a.  a„>0.  (1 


For, 

Hence 
As 


lim^=l, 


we  need  only  to  apply  177  to  2)  to  get  1). 

179.  Jjet  a^a^  -••  be  a  sequence  whose  limit  is  -{-  ao  .      Then 

limlog,a,=  |      Q     .^^^^^ 

Let  h>l.     Let  m^  be  an  integer,  such  that 

h"'n<a^<h"'n+^. 
Then,  by  176, 

log  a„  >  w„.  (1 

But 

lim  m„  =  +  Qo, 

since  lim  a„  =  +  oo. 

Hence,  by  138,  using  1), 

lim  log  an=  +  00. 

The  case  that  5  <  1  follows  at  once  now. 

Some  Theorems  on  Limits 

180.  Let  A  =  a  J,  «2'  "*  ^^  any  sequence^  such  however  that  its  limit 
is  ±30  when  it  is  7iot  limited;  let  B—h^^  h^,'--he  an  increasing 
seqxience  tvhose  limit  is  -{-  co.     If 


is  finite  or  infinite,  then 

lim  p  =  I.  (2 

On 


SOME   THEOREMS  ON   LIMITS  118 


Proof,    r.  I  finite. 
Set 

bn+i-b„  '"      b„^p-b^ 

From  1)  we  have  : 

S  >  0,     m,     \l  —  q„\<8,     n^m. 
Hence 

To  these  inequalities  apply  93,  setting  the  7's  equal  1.     Then 
or 

qm,p-B<i<qm,p  +  B'  (3 

*Tf  A  is  limited  ; 

since,  by  hypothesis,  5„  =  +  00. 

In  3,  pass  to  the  limit  jy  =  oo  ;   we  get 

-8<l<8. 
Hence 

1=0. 


But  on  the  supposition  that  A  is  limited, 

a, 
6. 


lim  ^  =  0. 


Thus  2)  holds  in  this  case. 

If  A  is  not  limited  ; 
8>Q  being  small  at  pleasure,  and  m  fixed,  we  have,  by  92, 

1 K. 

S>0^    i>o>      ^  =  1  +  Si;     \8^\<8,    p>p,.  (4 


''m+p 


U4 


EXPONENTIALS  AND  LOGARITHMS 

r 


a 


m+p 


Now 

Also,  by  3), 


Qm,p 


^m+p\  -*-  7 


From  5)  we  get,  using  4),  6), 


m+p^ 

3'i<a. 


^m+p 


Hence 


^  -  Z  =  Zg^  +  g' +  SjS', 


and 


supposing  h<l. 
If  now  we  take 

7)  gives  2), 


'm+p 


7 "'m+p 


'm+p 


<s(|;i  +  2),' 


s< 


2+ur 


2°.  Z  infinite.     To  fix  the  ideas,  suppose  ?  =  +  co. 
Then 

^>0,  w,  f^>^.         n>m. 
Hence 

Applying  93,  we  get 

qm,p>9-       ^  =  1,2,... 
This  shows  that 

lim  a„  =  +  00. 

Then  5)  shows  that,  taking  7}  such  that  0  <  77  <  1, 


9'»»,P  =  ^(1+V);   |V|<^,i'>i?o' 


by  92.     Hence 


^m.+p qm,p 


T> 


'm+p 


1  +  77'        1  +  77 


(5 
(6 


.0 


SOME   THEOREMS   ON   LIMITS  115 

If  we  take 

where  Cr  is  arbitrarily  large,  we  have 

-=^>Gr.         n>m+p. 
o„ 

n 

This  proves  2)  for  this  case. 

181.  Let  aj,  a^^  ■■■  be  a  sequence  whose  limit,  finite  or  infinite,  is 
a.      Then 

lim^i  +  ^^  +  --   +^-  =  a. 
n 

Let 

A  =  «!  +  •••  +  ««• 

Then 

Hence,  by  180, 

lim  -^  =  a. 

182.  Let  aj,  ag'  •"  ^^  ^  sequence  whose  terms  are  positive,  and 
whose  limit,  finite  or  infinite,  is  « >  0. 

Then  ^^ 

\\va.^a^a^---a^=  a.  (1 

1°.  a  finite.     Consider  the  auxiliary  sequence 

log  a,,  loga„,  •••;  base>l. 
By  178, 

lim  log  a„  =  log  a.  (2 

By  181  and  2), 

lim  -  (log  a.-\ h  log  a«)  =  log  a.  (3 

n 

But 

-(log  «!+•••+ log  a„)=l0g^«l«2-"^«-  (4 

n 
From  3),  4)  we  have  1). 
2°.  a  infinite.     To  fix  the  ideas,  let  a  =  +  oo. 

By  179, 

hm  log  a„  =  +  00. 


116  EXPONENTIALS   AND   LOGARITHMS 

By  181, 


Thus 


lim  log  Vai«2  •••(««  =  «=  +  CO. 


lim  Vaj  •••«„=  +  00. 
Hence  1)  is  true  in  tliis  case. 

183.    Let  aj,  a^,  •••  he  a  sequence  of  positive  numbers. 
Let 


Then 
For, 


lim  — ^  =  «  ^  0  ;  finite  or  infinite. 


Inn  V«,,  =  «. 


"  »   /^  /7  /»  '■ 


Apply  now  182. 


(^n-X       ^n-'l        "-1 


184.    Let  a  J,  a^.,  •■•  he  a  sequence  whose  limit  is  0. 
Let  5j,  621  ■••  he  a  decreasing  sequence  whose  limit  is  0. 
Let 


linir^^: 7^^^  =  I ;  finite  or  infinite. 


Then 


hn  -  ^«-l 


1°.  I  finite.     As  in  180,  we  have 


e>0,  m'. 
Since  by  92, 


^m        ^m+p 


"m  "rii+p 


<J;i^  =  i'2, 


m>m 


lim    '"■      "m+p ^TO 


p="  6, 


we  have 


^m+p         ^1)1 


€'   ^0' 


a^,  —  a 


m         "^nH-j>         ^«i 


^m  ^m+p         ^m 


€ 

<2' 


P>?o- 


Adding  2),  3),  we  get 


(1 


(2 


(3 


-I 


<  e ;  w  >  ?n'. 


SOME   THEOREMS   ON   LIMITS  117 


2°.   I  iyifinite.     Let  Z  =  +  oo 
Then 

«.„,.  —  a 


a>0,  m',  5„,^=-^!^L_^>^.  ^  =  1,2,...         m>m'. 


TO  ^m+p 

But  for  sufficiently  large  jd, 


Hence 


Hence 


^.«,p  =  |^.+  Si;  |Si|<S,  by  92. 


-~>g\         m>m'. 


and  ^  is  large  at  pleasure,  since  G  is. 

185.  EXAMPLES 

J  lim  logw  _  Q 


For, 


log  w- log  (n  -1)  ^  j^g     n      ^  jj^ 
71  —  (?i  —  1)  °  n  —  1 


2.  lime»^^^_ 

For, 


fLzurL=  en  (l -'-]=+ ^ 

I— (n  — 1)  \        ey 


3.  li™\/n=l. 

For, 

=  1. 


n-1 


4.  li™  v/n  !  =  +  00. 

For, 


•TS  C. 

X  ^ 

<  -i 

w-o 

o2 

W  o 

DO 

3^ 

CHAPTER   IV 

THE  ELEMENTARY  FUNCTIONS.     NOTION  OF  A  FUNCTION 
IN   GENERAL 

FUNCTIONS  OF   ONE   VARIABLE 

Definitions 

186.  The  functions  of  elementary  mathematics  are  the  following : 

Integral  rational  functions.  Exponential  functions. 

Rational  functions.  Inverse  circular  functions. 

Algebraic  functions.  Logarithmic  functions. 
Circular  functions. 

The  reader  is  already  familiar  with  the  simpler  properties  of 
these  functions,  which  we  may  call  the  elementary  functions.  We 
wish,  however,  to  restate  some  of  them  for  the  sake  of  clearness. 

187.  In  applied  mathematics  we  deal  with  a  great  variety  of 
quantities,  as  length,  area,  mass,  time,  energy,  electromotive  force, 
entropy,  etc.  In  a  given  problem  some  of  these  quantities  vary, 
others  are  fixed  or  constant. 

The  measures  of  these  quantities  are  numbers. 

In  certain  parts  of  pure  mathematics  we  study  the  relations 
between  certain  sets  of  numbers  without  reference  to  any  physical 
or  geometrical  quantities  they  may  measure.  In  either  case  we 
find  it  convenient  to  employ  certain  letters  or  symbols  to  which 
we  assign  one  or  more  numbers,  or  as  we  say,  numerical  values. 

A  symbol  which  has  only  one  value  in  a  given  problem  is  a 
constant. 

A  symbol  which  takes  on  more  than  one  value,  in  general  an 
infinity  of  values,  is  a  variable. 

118 


DEFINITIONS  119 

188.  The  set  of  values  a  variable  takes  on  is  called  the  domain 
of  the  variable. 

It  is  often  convenient  to  represent  the  values  of  a  variable  by 
points  on  a  right  line  called  the  axis  of  the  variable,  as  explained 
in  123.  The  domain  of  a  variable  may  embrace  all  the  numbers  in 
9?,  or,  as  is  more  often  the  case,  only  a  part  of  these  numbers.  Very 
frequently  the  domain  is,  speaking  geometrically,  an  interval ;  i.e. 
the  variable  x  takes  on  all  values  satisfying  the  relation 

a<x<h. 

Such  an  interval  we  shall  represent  by  the  symbol 

(a,  5). 

Frequently  one  or  both  the  end  points  a,  h  are  excluded. 
Then  we  use  the  symbols 

(^a*,b')  for  a<x<b; 

(a,  5*)   for  a<x<b; 

(a*,  6*)  for  a<x<b. 
Similarly, 

(a,  +  oo)  includes  all  x'^ai 

( —  00,  a)  includes  all  x^a; 

(—00,  +  oo)  includes  all  x  in  9?. 

A  point  of  the  interval  (a,  5)  which  is  not  an  end  point  is  within 
the  interval. 

189.  Let  a:  be  a  variable,  whose  domain  call  D.  Let  a  law  be 
given  which  assigns  for  each  value  of  x  in  2>  one  or  more  values 
to  ?/.     We  say  i/  is  a  function  of  x,  and  write 

^=/(2;)»  or  ^  =  </>(a^)'  etc. 

If  1/  has  only  one  value  assigned  to  it  for  each  value  of  x  in  D, 
we  say  ?/  is  a  one-valued  function,  otherwise  y  is  many-valued. 
The  variable  x  is  called  the  independent  variable  or  argument;  y  is 
called  the  dependent  variable. 


120    ELEMENTARY   FUNCTIONS.    NOTION   OF   A   FUNCTION 

We  must  note,  however,  that  y  may  be  a  constant. 

The  domain  of  the  independent  variable  x  which  enters  in  tlie 
law  defining  a  function  f{x)  is  also  called  the  domain  of  definition 
of  the  function. 

The  above  very  general  definition  of  a  function  is  due  to 
Dij'ichlet. 

190.    The  reader  is  already  familiar  with  the  graphical  repre- 
sentation of  a  function,  by  the  aid  of  two  rectangular  axes. 
Let  ^,  . 


be  a  given  function  whose  domain  of  defini- 
tion call  D. 

The  graphical  representation  of  i)  is  a  set      

of  points  on  the  x-axis. 

Let  a  be  a  value  of  x  to  which  corresponds 
the  value  b  of  y.     The  point  P  in  the  figure 
whose  coordinates  are  a,  h  represents  the  value  of  the  function 
for  x  =  a. 

As  x  runs  over  the  values  of  its  domain  2),  the  point  P  runs 
over  a  set  of  points,  which  we  call  the  graph  of  fix). 

191.  1.  Another  representation  of  a  function  is  the  following: 
We  take  two  axes  as  in  the 

figure  ;  one  for  x,  one  for  y.      x 1 

In    this    representation,    the 

graph  of /(a;)  is  a  set  of  points        i© J^ 

on  the  3/-axis. 

2.  The  reader  will  observe  this  important  difference  between 
the  two  modes  of  representation  just  given.  In  the  first  we  know 
for  each  point  P  of  the  graph  the  corresponding  values  of  both  x 
and  y.  In  the  second  mode  of  representation,  we  do  not  know 
in  general  the  value  of  x  corresponding  to  a  point  P  of  the 
graph,  and  conversely.  In  spite  of  this  deficiency,  we  shall  find 
that  this  second  representation  is  extremely  useful.  This  is 
especially  the  case  when  we  come  to  consider  functions  of  n 
variables. 


INTEGRAL   RATIONAL   FUNCTIONS 


121 


192.    Ex.  1.     Let  D  be  given  by 
0<X<1; 
while  y  is  given  by 

y  =  x^. 

The  graph  of  y  in  the  first  mode 
of  representation  is  the 'arc  of  the 
parabola  given  in  Fig.  1.  The  do- 
main of  definition  D  is  the  segment 
(0,  1)  on  the  rc-axis,  drawn  heavy  in 
the  figure. 

In  the  second  mode  of  representa- 
tion the  graph  of  y  is  the  segment  marked  heavy  on  the  ?/-axis 
(Fig.  2). 


Fig.  2. 


193.    Ex.  2.    As  in  Ex.  1,  let 


y^x^. 


Let,  however,  the  domain   of  definition  D  embrace  only  the 
values  of  ™_i     i    i    i     ... 

•*'  —  ^1    2'    3'    1' 


In  the  first  representation  the  graph  of  y 
is  a  set  of  points  lying  on  the  arc  of  the 
parabola  of  Ex.  1. 

In  the  second  representation  it  is  the  set 
of  points 

on  the  ?/-axis. 

In  both  modes  of  representation,  the  representation  of  D  is  the 
set  of  points 


1'     9'     16' 


\c,\ih    V2 


on  the  a;-axis. 


11111 

-^'    2'    3'    1'    "5' 


O       1111 


oTT 


Integral  Rational  Functions 
194.    These  functions  are  of  the  type 

?/  =  ao  +  a-^x  -f-  a^x"^  -|-  •  •  •  -f  a„,2;™, 
where  the  a's  are  constants,  and  w  is  a  positive  integer  or  0, 


(1 


122    ELEMENTARY   FUNCTIONS.     NOTION   OF   A   FUNCTION 

Such  functions  are  called  polynomials  in  algebra. 

The  number  m  is  called  the  degree  of  the  polynomial  y. 

When  m  =  1,  we  have 

y  =  a^  +  a^x.  (2 

The  graph  of  2)  is  a  straight  line.     For  this  reason  an  integral 
rational  function  of  the  first  degree  is  called  linear. 
When  w  =  0  or  when  a^  =0  in  2),  we  have 

y  —  ttf^^  a  constant.  (3 

We  still  say  ?/  is  a  function  of  x.  In  fact,  3)  states  that  for 
each  value  of  x,  the  corresponding  value  of  y  is  «(,. 

The  graph  of  3)  is  a  line  parallel  to  the  x-axis.  Since  the 
equation  1)  assigns  to  y  a  value  for  each  value  of  x^  the  domain  of 
definition  of  y  embraces  all  numbers  of  9?.  Speaking  geometrically, 
as  we  often  shall  in  the  future,  it  includes  all  the  points  of  the 
a;-axis. 

Since  1)  assigns  only  one  value  to  y  for  each  value  of  x^  y  \s  & 
one-valued  function. 

195.    In  algebra  we  learn  that  if  a  polynomial 

^m  =  «o  +  «ia;+  •••  a;„a;"'  (1 

vanishes  for  a;  =  a,  we  can  write 

where  P^-i  is  a  polynomial  of  degree  m  —  1.  We  learn  also  in 
algebra  that  a  polynomial  of  degree  m  cannot  vanish  more  than  m 
times,  without  being  identically  0,  in  which  case  all  the  coefficients 
in  1)  are  0. 

Should  1)  vanish  for 

a;=«j,  a^,  ■■■  a^,  (2 

we  have 

^m  =  «m(a^  -  «i)(a^  -  «2)  •••  (2^  -  O- 

The  numbers  2)  are  called  roots  or  zeros  of  P^. 


ALGEBRAIC   FUNCTIONS  123 

Rational  Functions 

196.  The  quotient  of  two  integral  rational  functions  of  x  is 
called  a  rational  function. 

Their  general  type  is  given  by 

a^  +  a^x+  —  -\-a^x"'  ^^ 

where  the  a's  and  5's  are  constants  and  m,  n  are  positive  integers 
or  0. 

The  expression  1)  involves  division  by  zero  for  those  values  of 
X  for  which  the  denominator  vanishes.  The  domain  of  definition 
2)  of  a  rational  function  includes  therefore  all  points  on  the  2;-axis 
except  the  zeros  of  the  denominator.  These  zeros  we  shall  call 
the  poles  of  ?/.  Since  1)  assigns  only  one  value  to  ?/  for  each  point 
of  D,  a  rational  function  of  x  is  one-valued. 

The  degree  of  7/  is  the  greater  of  the  two  integers  m,  n ;  suppos- 
ing, of  course,  that  a„„  b^^O. 

When  7/  is  of  the  first  degree,  it  is  called  linear.  The  type  of  a 
linear  rational  function  is,  therefore, 

gp  -f-  g^a; 
5o  +  h^x 

The  rational  function  includes  the  integral  rational  function  as 
a  special  case. 

In  fact,  let  the  numerator  be  divisible  by  the  denominator,  then 
1)  reduces  to  a  polynomial.  This  takes  place  in  particular  when 
the  denominator  reduces  to  a  constant. 


Algebraic  Functions 

197.    We  say  y  is  an  algebraic  function  of  x  when  it  satisfies  an 
equation  of  the  type 

2/"  +  R,y-'  -f  R^-'  -F-  . . .  -f  R,_^  +  i2.  =  0,  (1 

where  w  is  a  positive  integer,  and  the  i2's  are  rational  functions 
of  X. 

The  degree  oi  y  \&  n. 


124    ELEMENTARY   FUNCTIONS.     NOTION   OF   A   FUNCTION 

Let  us  give  to  a;  a  definite  value  x  =  a.  If  a  is  a  pole  of  any  of 
the  i^'s,  the  equation  1)  has  no  meaning  for  this  point,  and  it  does 
not  lie  in  the  domain  of  definition  of  y. 

Suppose  a  is  not  a  pole  of  any  of  the  i2's. 

Then  each  i?,^  takes  on  a  constant  value,  say  J.^,  and  1)  goes 

over  into  y.  + A,/- +  -  +  A_,y  +  ^.  =  0,  (2 

an  equation  with  constant  coefficients  of  degree  n. 

Equation  2)  may  have  no  real  roots.  In  this  case  a  does  not 
belong  to  the  domain  of  definition  of  y.  On  the  other  hand,  2) 
cannot  have  more  than  n  real  roots  for  x=^a. 

These  considerations  show  that  y  is  at  most  an  ?i-valued  function 
whose  domain  of  definition  embraces  all  or  only  a  part  of  the  2;-axis. 

If  we  clear  of  fractions  in  1),  we  may  write  it 

where  now  the  coefficients  of  y  are  polynomials  in  x.     Evidently 
either  1)  or  3)  may  be  used  as  definition  of  an  algebraic  function. 

198.  The  algebraic  functions  include  the  rational  functions  as 
a  special  case. 

For,  if  w  =  1,  the  equation  1)  of  197  takes  on  the  form 

or  y=-  Ry 

But  —  H^  is  any  rational  function. 

199.  An  expression  which  can  be  formed  from  x  and  certain 
constants  by  the  four  rational  operations  and  the  extraction  of 
roots,  each  repeated  a  finite  number  of  times,  is  called  an  explicit 
algebraic  function. 

Such  a  function  is 

Obviously  1)  can  be  obtained  from 

X,  a,  b,  c,  d 


CIRCULAR  FUNCTIONS  125 

by  the  aid  of  the  five  operations,  addition,  subtraction,  multiplica- 
tion, division,  and  the  extraction  of  roots,  each  repeated  only  a 
finite  number  of  times. 

It  is  known  that  every  explicit  algebraic  function  y  satisfies 
an  equation  of  the  type  197,  1).  Hence  every  explicit  algebraic 
function  is  an  algebraic  function,  by  197. 

The  converse  is,  however,  not  true ;  every  algebraic  function  y 
cannot  be  brought  into  the  form  of  an  explicit  algebraic  function. 

This  is  due  to  the  fact  that  equations  of  degree  w>4  cannot,  in 
general,  be  solved  by  the  extraction  of  roots,  or,  as  we  say,  do  not 
admit  of  an  algebraic  solution. 

200.  All  fuiictions  which  are  not  algebraic  functions  are  called 
transcendental  functions. 

The  terms  algebraic  and  transcendental  may  also  be  applied  to 
the  numbers  of  9?. 

Any  number  a  which  satisfies  an  equation  of  the  type 

a;"  +  a^x""-^  +  a^x"'"^  -\ \-  a^_-^x  +  a„  =  0,  (1 

where  w  is  a  positive  integer,  and  the  a's  are  rational  numbers,  is 
called  an  algebraic  number.  All  other  numbers  of  9^  are  transcen- 
dental numbers. 

When  w  =  1,  the  equation  1)  defines  a  rational  number ;  the 
rational  numbers  are  special  cases  of  algebraic  numbers. 

Circular  Functions 

201.  As  the  reader  already  knows,  the  circular  functions  may 
be   defined    as    the   lengths    of    certain   lines 
connected  with  a  circle  of  unit  radius. 

Thus,  in  the  figure 

sin  x  =  CE,  cos  x  —  OE,  tan  x  =  AB, 

etc.  We  have  shown  in  Chapter  II  how  the 
rectilinear  segments  AB,  CE,  etc.,  are  meas- 
ured. It  has  not  yet  been  shown,  however, 
how  to  measure  arcs  of  a  circle,  i.e.  how  to 
each  arc  as  AC,  a  number  x  may  be  attached,  as  its  measure. 


126     ELEMENTARY   FUNCTIONS.     NOTION   OF   A   FUNCTION 

This  will  be  given  later.  No  inconvenience  can  arise  if  we  assume 
here  a  knowledge  of  this  theory  inasmuch  as  the  reader  is  perfectly 
conversant  with  its  results,  which  are  all  we  need  for  the  present. 

Arcs  measured  in  the  direction  of  the  arrow  are  positive ;  those 
measured  in  the  opposite  direction  are  negative. 

If  we  suppose  the  point  0  to  move  around  the  circle  in  a  positive 
direction  starting  from  a  fixed  point  A  as  point  of  reference,  it  has 
described  an  arc  whose  measure  is  2  tt, 

7r=  3.14159265... 

when  it  reaches  A  again.  If  it  still  continues  moving  around  the 
circle,  it  has  described  an  arc  =  4  tt  when  it  reaches  A  for  the 
second  time.  On  reaching  A  for  the  third  time  the  arc  described 
is  6  TT,  etc.  Thus  to  each  positive  number  in  9?  corresponds  an 
arc;  also,  conversely,  to  each  arc  measured  in  the  direction  of  the 
arrow  corresponds  a  positive  number  in  9?. 

With  arcs  measured  in  the  negative  direction  are  associated  the 
negative  numbers  of  9?,  and  conversely. 

202.  From  this  mode  of  defining  the  circular  functions  we  con- 
clude at  once  the  following  properties  : 

The  domain  of  definition  of  sin  a;,  cos  x  embraces  all  the  numbers 
of  9ff. 

The  domain  of  definition  of  tan  x  embraces  all  numbers  of  9? 
except 

5  +  ^^'  (1 

where  m  =  0,  ±1,  ±2,  •••. 

In  fact,  for  these  arcs,  the  secant  OB  is  parallel  to  the  tangent 
line  AB,  and  therefore  cuts  off  no  segment  on  it.  Thus  for  these 
values  of  the  argument  a;,  tan  x  is  not  defined. 

Similarly,  sec  x  is  not  defined  for  these  same  values  1) ;  while 
cosec  X  is  not  defined  for 

X  =17117.  w=  0,  ±  1,  ±  2,  •••  (2 

From  similar  triangles  we  have  for  all  x,  except  these  singular 
values  in  1)  oy  2), 

sin  X                    1  1 

tana:= -,  seca;= ,  cosec  2;  = 


cos  z  cos  X  sin  x 


CIRCULAR   FUNCTIONS 


127 


We  observe  that  these  relations  involve  division  by  0,  for  the 
singular  values  1)  or  2). 

From  the  above  definition  of  the  circular  functions  we  see  that 
they  are  one-valued  functions  of  x. 

203.  The  graphs  of  the  three  principal  functions  sin  a:,  cos  a;, 
tana;,  are  given  below. 


\^ 

A 

y 

J 

A                  J 

A 

X 

.Z IT  Air    -IT 
-Y/            2 

\ 

0     f 

tanx 

Sir 
2 

204.  The  next  most  important  property  of  the  circular  function 
is  their  periodicity . 

In  general  we  define  thus : 

Let  &)  be  a  constant  t^O.  Let  f(x)  be  a  one-valued  function 
whose  domain  of  definition  D  is  such  that,  if  x  is  any  point  of  2), 

so      is  -,  r. 

x+mo),         m=±l,    ±2,   ••• 

^^  /(^  +  «)=/(^)  (1 

for  every  x  in  D,  we  say  /(a;)  is  periodic,  and  admits  the  period  to. 
If  6)  is  a  period  of  /(a;),  so  is  mco. 

m  =  ±l,   ±2,  ... 
For, 

fCx  +  2  to)  =/[(a;  +  a,)  +  «]  =f{x  +  a,)  =/(a:), 

^  hence  2 co  is  a,  period.     Similarly,  3©,  4  to,  ...  are  periods. 
Qn  the  other  hand, 

/(a;  =  0))  = /((a;  -  «)  +  a>)  = /(a:). 


128    ELEMENTARY   FUNCTIONS.     NOTION   OF   A   FUNCTION 

Hence  f(x)  admits  the  period  —  to,  and  so  —2  ft),  —3  ft),  etc. 
If  all  the  periods  that  f{x)  admits  are  multiples  of  a  certain 
period  S,  this  is  the  primitive  period  of /(a;),  or  the  period  oi  f(x). 
From  trigonometry  we  have : 

The  period  of  sin  x^  cos  a;,  is  2  tt  ;  the  period  of  tan  x  is  ir. 

205.    1.  If  f{x)^  g(x)  admit  the  period  &),  then 

f(ix-)±g(ix\  (1 

fix-)g<ix),    4^,    ^(^)^0„  (2 

admit  also  the  period  co. 
For  example,  let 

We  have 

h(x  +  ft))  =  fix  +  ft))  +  g(x  -r  ft))  =f(x')  +  g(x}  =  h(x'). 

2.  If  the  period  of  fQc)  is  ft),  the  period  of  f(ax^  is  —.     Here  a 
is  ariy  number  ^  0. 

5'(^)=/(«2J)> 

and  let  r  be  any  period  of  g(x). 

Then 

g{x  +  T)=g(x^, 

''''  /«a:  +  T))  =/(«:.). 

This  gives,  setting  _ 

/0  +  aT)=/(0. 
Thus  ar  is  a  period  of  f(x) ;  and  therefore 

ar  =  mat.         m  an  integer. 

Hence 

T  =  m— .  (3 

a 

As  -  is  obviously  a  period  of  ^(a;),  it  is  *Ae  period  of  g(x),  by  3), 


CIRCULAR  FUNCTIONS  129 

3.  Suppose  in  1,  that  &>,  instead  of  being  any  period  of  /  and  g^ 
is  the  period  of  these  functions.  It  is  important  to  note  that  we 
cannot  infer  that  therefore  a>  is  the  period  of  the  functions  in  1),  2). 

Ex.  1.   Let 

fix)  =  sin  X,    g(x)  =  4  sin  x  cos'^  x. 

The  period  of  these  functions  is  2  ir. 
Yet  the  period  of 

h(x)  =  g{x)  —f(x)  =  sin  3  a; 


is  2^ 
3 


Ex.  2.    Let 

f{x)  —  sin  x,   g(x)  =  cos  x. 
Then 

h{x)  ~f{x)g{x)  =  i  sin  2  x. 

The  period  of  /  and  gr  is  2  tt  ;  the  period  of  h  is  ir. 

Ex.  3.    Let  /(x),  </(x),  be  as  in  Ex.  2.     Let 


The  period  of  A  is  again  ir. 
Ex.  4.    Let 


A(x)  =  4^  =  tana;. 


f(x)  =  sin'-  X,    g{x)  =  cos*  x. 

The  period  of  /,  g  is  tt. 

But 

h(x)^f(x)  +  g{x)  =  l, 

which  has  no  primitive  period. 

206.    From  the  periodicity  of  the  circular  functions  we   can 
prove  that 

The  circular  Junctions  are  transcendental. 

Consider,  for  example, 

y  =  sin  X. 

If  this  is  algebraic,  let  it  satisfy  the  irreducible  equation 

y-  +  R^ix)y-'  +  -+R„ix')^^.     ,  ^ 

Replace  here  a;  by  a;  +  2?7Z7r,  m  an  integer. 

If  we  set 

R^{x+2mir)=T^{x\ 

1)  gives,  since  y  is  unaltered, 

y-  +  T^ix)y--'  4-  -  +  T,(x)  =  0.  (2 


130    ELEMENTARY  FUNCTIONS.    NOTION  OF   A  FUNCTION 

As  y  satisfies  both  1)  and  2),  it  satisfies  their  difference 
(i^i  -  7\)^"-i  +  ...  +  (i2„  -  ^„)  =  0. 

Thus   y   satisfies  an  equation  of  degree  <  ti,  which   is   not   an 
identity.     As  we  assumed  1)  is  irreducible,  this  is  a  contradiction. 

207.    Another  important  property  of  the  circular  function  is 
their  addition  theorem,  which  is  expressed  in  the  formulae 

sin  (x  +  y^  =  sin  x  cos  y  +  cos  x  sin  ^/, 

cos  (x  +  y')  =  cos  X  cos  y  —  sin  x  sin  y. 


etc. 


tan  X  +  tan  y 

tan  (x->ry)=  z. — =^» 

1  —  tan  X  tan  y 


208.    1.   Let  fipc)  be  a  one-valued  function  whose  domain  of 
definition,  i>,  is  such  that  if  x  is  any  point  of  D,  so  is  —  x. 

Let  /(-a:)=/(a;), 

for  every  x  in  D.     We  say,  then,  that /(a;)  is  an  even  function. 
If 

we  sa,jf(x')  is  an  odd  function. 
Obviously, 

The  functions  sin  x,  tan  x  are  odd,  while  cos  x  is  even. 

2.  Letting  0,  0^,  0^  represent  odd  functions,  and  E,  E^,  E^  even 
functions,  we  have: 

0±0^  =  0,^,     E±E,  =  E^, 
0'0^  =  E^^,     EE^  =  E^,     0'E=0^, 

For  example : 


ONE-VALUED   INVERSE   FUNCTIONS 


131 


,   The  Exponential  Functions 

209.    Let  a  >  0  be  a  constant ;   the  exponential  functions  are 

defined  by 

y  =  w'. 

The  domain  of  definition  of  y  is  9^?,   and  ^  is  a  one -valued 

function. 

When  a  =  1,  the  corresponding  exponential  function  reduces  to 

a  constant,  viz. : 

y  =  l. 

The  graphs  of  y  fall  into  two  classes,  according  as  a^l. 
y 


An   important   exponential   function  is  that  corresponding  to 
^^^'  e  =  2.71818 - 

210.    1.  The  only  properties  of  the  exponential  functions  which 
we  care  to  note  now  are  the  following  : 

The  exponential  function  is  noivhere  equal  to  0,  or  any  negative 
number. 
See  165,  2. 

2.  The  addition  theorem  is  expressed  by 


One-valued  Inverse  Functions 

211.  1.  The  two  remaining  classes  of  functions,  viz.  the  logarith- 
mic and  inverse  circular  functions,  are  inverse  functions.  Before 
considering  them,  we  wish  to  develop  the  notion  of  inverse  func- 
tions in  general. 


132    ELEMENTARY   PUXCTIONS.    NOTION   OF   A  FUNCTION 

2.  Let /(a;)  be  a  one-valued  function  defined  over  a  domain  D.* 

If  fix")>f(x') 

for  every  pair  of  values  x"  >  x'  in  i),  we  say  f(j>c)  is  an  increasipg 
function  in  D. 

If  on  the  contrary 

f(x")<f(ix'\         2;">a/, 

we  say/(a:)  is  a  decreasing  function. 

If/ is  either  an  increasing  or  a  decreasing  function  in  7),  but  we 
do  not  care  to  specif}^  which,  we  say  it  is  univariant. 

These  definitions  are  extensions  of  those  given  in  108.  Tlie 
corresponding  extension  of  the  terms  monotone,  monotone  increas- 
ing, monotone  decreasing  to  function  is  obvious. 

3.  Ex.  1.  For  the  domain  D  =  (  —  -  ,    ~  ] ,  sin  a^  is  an  increasing  function. 
For  the  domain  Z)  =  [ -,    — ^ ),  si"  a;  is  a  decreasing  function. 

Ex.  2.  For  the  domain  3  =  ^^,0^  is  an  increasing  function  if  a  >  1  ;  it  is  a  de- 
creasing function  if  a <  1.     Thus  whether  a^\,  a^  is  a  univariant  function  in  1R. 

212.  Let  „ ,  .  ^. 

y=f(.x)  (1 

be  a  one-valued  univariant  function,  defined  over  a  domain  D. 
Let  E  be  the  domain  over  which  the  variable  y  ranges. 

We  put  the  points  of  D  and  E  in  correspondence  with  each 
other  as  follows :  two  points  x,  y  shall  correspond  to  each  other, 
or  be  associated,  when  they  satisfy  1). 

Then  to  a  given  x  corresponds  only  one  y,  since  /(a:)  is  one- 
valued.  On  the  other  hand,  to  a  given  y  corresponds  only  one  x, 
since /(a;)  is  univariant. 

Thus  to  any  x  oi  D  corresponds  one,  and  only  one,  y  oi  E  -, 
conversely,  to  any  y  oi  E  corresponds  one,  and  only  one,  x  of  D. 

213.  The  considerations  of  the  last  article  have  led  us  to  one 
of  the  most  important  notions  of  modern  mathematics,  that  of 
correspondence. 

*  Such  an  expression  as  this  will  be  constantly  employed  in  the  future.  It  does  not 
mean  that  D  includes  all  the  values  for  which /(a;)  may  be  defined,  but  only  such  values 
as  one  chooses  to  consider  for  the  moment. 


ONE-VALUED   INVERSE   FUNCTIONS  133 

Let  A  and  B  be  two  sets  of  objects.  Let  us  suppose  that  A 
and  B  stand  in  such  a  relation  to  each  other,  that  to  any  object 
a  oi  A  correspond  certain  objects  6,  b',  b",  •••  of  B;  and  to  any 
object  b  oi  B  correspond  certain  objects  a,  a',  «",•••  of  A. 

Then  A  and  B  are  said  to  be  in  correspondence. 

If  to  each  a  corresponds  only  one  b,  and  conversely,  the  corre- 
spondence is  one  to  one  (1  to  1),  or  uniform. 

If  to  each  a  correspond  m  objects  of  B,  and  to  each  b  correspond 
n  objects  of  A^  the  correspondence  is  m  to  n. 

In  many  cases,  to  each  element  of  A  correspond  an  infinity  of 
objects  of  B,  or  conversely. 

214.  Let  us  return  to  212.  The  correspondence  we  established 
between  the  points  of  J)  and  ^  is  uniform.  This  fact  may  be  used 
to  define  a  one-valued  function  ^(^),  over  the  domain  U.  In  fact, 
let  X  correspond  to  i/.  Then  ^(y)  shall  have  the  value  x,  at  the 
point  y.     Then  ^  . 

The  function  (/,  just  defined,  is  called  the  inverse  function  of  f. 
Evidently  </  is  a   one-valued   function   in   _£'.      It   is   also   uni- 
variant. 

In  fact,  to  fix  the  ideas,  suppose /is  an  increasing  function. 
Then,  if 


we  have  >    -    it 


x'  <  x'\ 

y'  <y' 


Suppose  now  g  were  not  an  increasing  function.     Then  for  at 
least  one  pair  of  points, 

y'<y".  (1 


we  would  have 


x'  >  x" . 


We  cannot  have  x'  =  x"  ;  for  then  y'=y'\  which  contradicts 
1).  We  cannot  have  x'  >  x"  ;  for  then  y'  >  y" ,  which  again  con- 
tradicts 1). 

We  haA^e  thus  the  theorem  : 

Let  y  =^  f{x)  be  a  one-valued  univariant  function.,  defined  over  a 
domain  B.  Bet  E  be  the  domain  of  the  variable  y.  Then  the 
inverse  function.,  x  =  g(^y),  is  one-valued  and  univariant  in  B. 


134    ELEMENTARY   FUNCTIONS.     NOTION   OF   A   FUNCTION 

215.  The  notion  of  inverse  functions  developed  in  212  and  214 
is  quite  general.  It  will  perhaps  assist  the  reader  if  we  take  a 
very  simple  case. 

For  the  domain  D  let  us  ^ 

take  an  interval  /=  (a,  J). 
For  /(a;),  let  us  take  an 
increasing  function,  with 
graph  as  in  the  figure.  The 
domain  of  y  is  then  the  in- 
terval J=(a^  yS).  That  the 
correspondence  between  the 
points  of  I  and  J,  as  defined 
in  212,  is  uniform,  is  seen 
here  at  once.  For,  to  find 
the  points  y  corresponding  to  a  given  a?,  we  erect  the  ordinate  at 
X.  This  cuts  the  graph  but  once,  viz.  at  P.  There  is  thus  but 
one  point  y  hi  J  corresponding  to  the  point  x  in  /. 

Similarly,  to  find  the  points  x^  corresponding  to  a  given  y,  we 
draw  the  abscissa  through  y.  This  cuts  the  graph  but  once, 
viz.  at  P. 

There  is  thus  but  one  point  x  in  /corresponding  to  a  given  y  in  J. 

That  the  inverse  function  is  one-valued,  and  is  an  increasing 
function  in  J,  is  at  once  evident  from  the  figure. 


The  Logarithmic  Functions 

216.    1.  We  saw  in  209  that  the  exponential  functions 

y=  a^,   a>0,  T^l 

are   one-valued   univariant   functions  for   the   domain  '?R..      The 
domain  of  the  variable  y  is  the  interval  1=  (0*,  +  oo).     See  188. 

Then,  by  214,  the  inverse  of  the  exponential  functions  are  one- 
valued  univariant  functions,  defined  over  /.     By  174,  these  inverse 

functions  are 

a:  =  log„^,  (1 

and  are  called  logarithmic  functions  with  base  a.      In  higher 
mathematics  it  is  customary  to  take  a  =  e=  2.71818  •••     When 


MANY-VALUED   INVERSE   FUNCTIONS 


135 


no  ambiguity  can   arise,   we   may  drop   the   subscript  a  in  1). 
Unless  otherAvise  stated,  we  shall  suppose  the  base  is  e. 

2.  The  graph  of  the  logarithmic 
function 

y=\ogx 

is  given  in  the  figure. 

3.  The  only  other  property  of  log  a; 
which  we  wish  now  to  mention  is  their 
addition  theorem, 

log  xy  =  log  X  +  log  y. 


Many-valued  Inverse  Functions 

217.  The  circular  functions  give  rise  to  many-valued  inverse 
functions.  It  is  easy  to  extend  the  considerations  of  212  and  214 
so  as  to  arrive  at  the  notion  of  many-valued  inverse  functions  in  all 
its  generality. 

Let 

2/=/(^)  (1 

be  a  one  or  many  valued  function,  defined  over  a  domain  D.  Let 
the  domain  of  the  variable  y  be  E.  We  put  the  points  of  D  and 
E  in  correspondence  as  follows  :  two  points  a;,  y  shall  correspond 
to  each  other  or  be  associated  when  they  satisfy  1).  Then,  to 
each  y  oi  E  correspond  one  or  more  values  of  a;,  say 


(2 


We  define  now  a  function  y(^y}  over  E  by  assigning  to  y  the 
values  2)  of  x  associated  with  each  point  y  of  E. 
Then 

^  =  9W  (3 

is  the  inverse  function,  defined  by  1). 

The  equation  3)  may  be  considered  as  the  solution  of  1)  with 
respect  to  x. 


136    ELEMENTARY   FUNCTIONS.     NOTION   OF   A   FUNCTION 


/s 

y 

j^ 

— - — n 

y 

-K"fe 

a 

"TT"    i" 

i      i    :  « 

0 

ax      X' 

X"  X"'b 

218.    To  illustrate  the  rather  abstract  considerations  of  the  last 
article,  let  us  consider  the  following  simple 
case,  from  a  geometric  standpoint. 

Let  the  graph  of 

y  =/(^) 
be  that  in  the  figure. 

Then  D  =  (a,  5),  and  U=  («,  yS). 

The  greatest  number  of  values  of  y  for  a  given  x  in  i>  is  3. 
Hence  ^  is  a  three-valued  function.  Let  y  be  a  point  of  -27.  To 
find  the  points  of  D  associated  with  it,  we  draw  the  abscissa 
through  ?/.  Let  it  cut  the  curve  in  the  points  P,  P',  P",  ••• 
The  projections  x,  x\  x",  ■■•  of  these  points  P  on  the  a;-axis  are 
the  points  sought. 

The  greatest  number  of  values  x  corresponding  to  any  y  ot  E 
is  4.     Hence  the  inverse  function 


is  a  four-valued  function. 


^=9iy) 


219.    Let  us  consider  the  function  y=f(x)  defined  by 


or 


y 


Its  graph,  given  in  the  figure,  is  a 
hyperbola. 

To  a  value  of  x>\^  or  x<  —  \^  correspond  two  values  of  y, 
marked  y  and  y'  in  the  figure.  The  domain  2>  of  a:  is  marked 
heavy  in  the  figure,  and  embraces  all  the  points  of  the  a;-axis, 
except  (—  1*,  1*).     The  domain  E  of  y  is  the  whole  ^-axis. 

To  any  point  y  oi  E  correspond  two  values  of  x,  falling  in  D. 

The  inverse  function  x  =  g{if)^  thus  defined,  is  a  solution  of  1) 
or  2)  with  respect  to  a;,  viz.  : 

x  =  ±  Vl  +  y"^. 

■  The   correspondence   which   the    equations  1)   or  2)   establish 
between  the  points  of  D  and  ^  is  a  2  to  2  correspondence. 


THE   INVERSE   CIRCULAR  FUNCTIONS  137 

220.  The  preceding  example  illustrates  the  fact  that 

The  inverse  of  an  algebraic  function  which  is  not  a  constant  is 
an  algebraic  function. 

To  prove  this  theorem,  let  y=f(x^  be  defined  by 

P,(x)y-  +P,(x)r~'  +   -  +^n(^)=  0,  (1 

where  the  P's  are  polynomials  in  x,  with  constant  coefficients. 
The  inverse  function 

^=9i:y)  (2 

also  satisfies  1).     Let  us  arrange  1)  with  respect  to  x.     If  m  is 
the  highest  degree  of  x  in  this  equation,  we  get 

^o(i/)^™  +  ^i(y)^™-'  +  -  +  Qmiy^  =  0.  (3 

As  2)  satisfies  3),  the  inverse  function  2)  is  an  algebraic  func- 
tion also. 

In  this  example,  ?/=/"(.r)  is,  in  general^  an  w- valued  function, 
while  x  =  g{jy)  is  an  w-valued  function. 

The  correspondence  that  the  equation  1)  or  3)  establishes  be- 
tween the  points  of  D  and  ^,  the  domains  of  the  variables  a;,  y,  is 
thus  an  n  to  m  correspondence. 

The  Inverse  Circular  Functions 

221.  These  are  the  functions 

sin~^a;,  cos"^a;,  tan^^a;,  etc. 

We  prefer  to  follow  continental  usage,  and  denote  them  respec- 
tively by 

Arc  sin  a:,  Arc  cos  a;,  Arctga;,  etc. 

We  shall  not  take  the  space  needed  to  treat  all  these  functions ; 
we  take  one  of  them.  Arc  sin,  as  an  illustration.  The  others  may 
be  treated  in  the  same  way. 

We  start  with  the  equation 

y  —  sin  a;,  (1 

whose  graph  is  given  in  203. 


138    ELEMENTARY   FUNCTIONS.     NOTION   OF   A   FUNCTION 
The  domain  i),  over  which  sin  x  is  defined,  is  9? ;  the  domain  of 

Let  ?/  be  a  point  of  M.     If  x^  is  one  of  the  associated  points  of 
Z>,  all  the  points  of  D  associated  with  y  are  given  by 

Xq+  2  mir,  (2 

m=0,   ±1,   ±2  ... 

TT—  Xq-\-2  WITT,  (3 

as  is  shown  in  trigonometry. 

Thus,  to  a  given  value  of  y  there  are  a  double  infinity  of  values 
of  X. 

The  inverse  function  defined  by  1),  viz, : 

X  =  Arc  sin  y, 

has  the  interval  ^  =  (—  1,  1)  for  its  domain  of  definition.  It  is 
an  infinite-valued  function  whose  values  for  a  given  y  are  given 
in  2),  3). 


222.    The  graph  of 


y  =  Arc  sin  a; 


is  given  in  the  adjoining  figure. 

The  reader  will  observe  that  this  graph  can  be  got  at 
once  from  the  graph  of  sin  x  (see  203)  by  turning  it 
around  and  changing  the  axes. 

This  property  is  obviously  true  of  the  graph  of  any 
inverse  function. 

Thus,  if  the  graphs  of 

e*,  cos  a;,  tana;,  etc., 
are  given,  we  may  get  at  once  the  graphs  of 

log  X,  Arc  cos  X,  Arc  tg  x,  etc. 


223.  The  treatment  of  many- valued  functions  is  much  simplified 
by  employing  the  notion  of  a  branch  of  the  function.  This  will 
be  explained  when  we  have  considered  the  notion  of  continuity. 
For  the  present,  however,  we  wish  to  define  what  are  called  the 
principal  branches  of  the  inverse  circular  functions. 


I 


THE  RATIONAL  AND  ALGEBRAIC  FUNCTIONS 


139 


Looking  at  the  graph  of  Arc  sin  x  given  in  222,  we  see  we  can 
define    a    one-valued   function     over     the 
interval  (—1,  1)  by  taking  those  values 
of   Arc  sin  x   which   fall   in   the   interval 
/_7r    7r\ 
l~2'  2J' 

The  function  so  defined  is  called  the 
principal  branch  of  the  Arcsin  futiction. 
We  shall  denote  it  by  arc  sin  x. 

Its  graph  is  given  in  the  adjoining 
figure. 


224.  1.  The  principal  branch  of  Arc  cos  x 
is  formed  of  those  values  of  this  function 
which  fall  in  the  interval  (0,  tt). 

The  one-valued  function  so  defined 
over  the  interval  (—1,  1)  is  denoted  by 
arc  cos  x. 

Its  graph  is  given  in  Fig.  1. 

2.  The  princii^al  branch  of  Arc  tg  x  is 
formed  of  those   values  of   this  function 

which  fall  in  the  interval  (  —  77,  -r )  • 

The  one- valued  function  so  defined  over  (  —  00,  oo)  is  denoted  by 
arc  tg  X. 


0        +1 

arc  cos  x 


Fig.  2. 
Its  graph  is  given  in  Fig.  2. 


IT 

2" 
00 

/-"^"^ 

00 

_7r 
T 

arc  tg  z 


FUNCTIONS   OF  SEVERAL  VARIABLES 

The  Rational  and  Algebraic  Functions 

225.  In  the  list  of  the  elementary  functions  given  in  186,  the 
first  three,  viz.  the  integral  rational,  the  rational,  and  the  alge- 
braic functions,  are,  in  general,  functions  of  several  variables. 


140    ELEMENTARY   FUNCTIONS.     NOTION   OF   A    FUNCTION 

For  simplicity,  we  treated  them  first  as  functions  of  a  single  varia- 
ble. We  wish  now  to  define  them  in  all  their  generality.  At 
the  same  time  we  shall  consider  the  general  notion  of  functions  of 
several  variables  and  certain  related  geometric  ideas. 

226.    1.   An  integral  rational  function  of  n  variables  x^,  x^  •••  Xn 
is  an  expression  of  the  type 

y  =  Ax^^x^-i  •  •  ■  a;„'""  +  Bx^^x^^  •  ■  ■  xjn  +  •  •  •  +  Lx^-^-x^-^  ■  ■  •  a;„*».     (1 

Here  AyB^-L  are  constants,  and  the  exponents  m,  Z,  •  •  •  e  are 
positive  integers  or  0.     Such  functions  are 

ax^x^x^  +  bx^x^  +  cx^  +  dx^'x^Xy  (3 

We  may  write  1)  in  the  form 

where  the  summation  extends  over  all  the  terms  of  y. 
A  still  shorter  notation  is 

y  =  ^Ax{"^x^''^-  ■  ■  ■  a:„,'"«  (5 

which  may  be  employed  when  no  ambiguity  can  arise. 
The  greatest  of  all  the  sums  of  the  exponents 

m^  +  7n^-\ h  nin,     li  +  l^'^ 1-  ^« ••' 

is  the  degree  of  y. 

Thus  the  degree  of  2)  is  3 ;  the  degree  of  3)  is  13. 

2.   When  the  degree  of  each  term  of  1)  is  the  same,  it  is  said  to 
be  homogeneous. 

F=^Ax{''x^'"'---x,,'^n 

he  homogeneous  and  of  degree  m.  If  in  F  we  replace  x^  by  \x^,  ••• 
x„  by  Xx„,  and  denote  the  result  by  F,  we  have 

F=  X^'F. 
For,  _ 

F=  '2A\"''^-^"'"x^"''  •••  x/^n. 


THE   RATIONAL   AND   ALGEBRAIC   FUNCTIONS  141 

But,  for  all  the  terms  of  F, 

*^i  +  •  •  •  +  wz„  =  Wi. 
Hence  _ 

F=  V^Ax{^t  •••  x^"n  =  \'"F. 

3.  When  3/  is  of  degree  1,  we  have 

y  =  aj^x^  4-  ^2X2  H 1-  a„x„  +  a^. 

It  is  said  to  be  a  linear  integral  function  of  the  rr's.  If  aQ=  0,  it 
becomes 

^  =  a^X^  +  •  •  •  +  CLnPCiii 

which  is  the  general  type  of  a  linear  homogeneous  integral  function 
of  the  x's,. 

In  algebra,  integral  rational  functions  are  called  polynomials. 

227.    1.  To  get  a  value  of 

y  =  l.Ax{''---x,,"'n  (1 

we  give  to  each  of  the  variables  x  a  certain  numerical  value,  as, 

X-^  =  a^i     ^2  ^^  '^2^        "     "^w  ^^  ^re*  \ 

These  values  put  in  1}  give  the  corresponding  value  of  ?/,  say 
y  =  h. 

When  w  =  1,  2,  3,  we  can  represent  geometrically  the  values  2) 
by  a  point  on  a  right  line,  a  point  in  a  plane,  or  a  point  in  space, 
respectively,  viz.  the  point  a  whose  coordinates  are  a^,  or  a^,  a^, 
or  a^,  a2,  a^  If  we  give  the  a;'s  different  sets  of  values,  we  get 
different  points  in  1,  2,  or  3  dimensional  space.  As  in  the  case 
of  one  variable,  we  can  say  y  has  the  value  h  at  the  point  a. 

2.  It  is  convenient  to  extend  these  and  other  geometric  terms, 
employed  when  the  number  of  variables  n  =  1,  2,  3,  to  the  case 
when  w>3.  Thus  any  complex  of  n  numbers,  a^,  a^.,  •••  a„,  is 
called  a  point;  a^,  a^.,  •••  are  called  its  coordinates.  We  denote 
the  point  by  _  ^  .  . 

The  aggregate  of  all  possible  points,  the  a;'s  running  over  all 
the  numbers  in  9?,  we  call  an  n-dimensional  space  or  an  n-way 
space;  and  denote  it  by  9?^.  Later  we  shall  extend  the  terms 
distance,  sphere,  cube,  etc.,  to  9?„.      Cf.  244.     The  reader  is  not  to 


142    ELEMENTARY   FUNCTIONS.     NOTION   OF   A   FUNCTION 

suppose  for  a  moment  that  there  really  is  an  7i-dimensional  space, 
or  an  w-dimensional  cube,  in  the  ordinary  empirical  sense  of  the 
word ;  but  to  bear  in  mind  that  these  terms  are  merely  names  for 
certain  numerical  aggregates. 

3.  Employing  this  geometrical  language,  we  may  say  that, 
The  integral  rational  function  of  several  variables,  say  n  variables, 

is  a  one-valued  function  whose  domain  of  definition  embraces  all  the 

points  of  9t„. 

228.  As  in  the  case  of  one  variable,  the  rational  function  of 
several  variables  is  the  quotient  of  two  integral  rational  functions 
in  these  variables.     Its  general  expression  is,  therefore, 

Its  domain  of  definition  embraces  all  the  points  of  9^?^,  except  those 
points  at  ivhich  Gr  vanishes,  which  we  call  poles  of  R.  For  all  points 
of  this  domain,  R  is  a  one -valued  function. 

If  m'  is  the  degree  of  F,  and  m"  is  that  of  Gr,  the  degree  of  R 
is  the  greater  of  the  two  integers  m' ,  m" . 

When  the  degree  of  R  is  1,  it  is  called  a  linear  rational  function. 
Its  general  expression  is 

flfq  -j-  a^x^  -|-  . . .  -|-  a^Xj^_  ,^2 

We  say  R  is  homogeneous  when  F  and  Gr  are  homogeneous. 
We  have  evidently,  as  in  226,  2, 

R(\X^,  \X^,   •••  \x^  =  \^R(x^  '"  Xn)j  (3 

where  t  is  an  integer,  positive,  negative,  or  zero. 

229.  The  definition  of  an  algebraic  function  of  n  variables  is  an 
obvious  extension  of  that  given  for  one  variable,  in  197.  Thus 
g  is  an  algebraic  function  of  x^,  x^,  •••  x^,  when  it  satisfies  an  equa- 
tion of  the  type 

g-  +  i^ir "'  +  •  •  •  +  ^n-iy  +  i^.  =  0,  (1 

where  the  coefficients  R  are  rational  functions  of  x^-^-x^,  and  n  is 
a  positive  integer. 


FUNCTIONS   OF   SEVERAL   VARIABLES   IN   GENERAL      143 

For  any  point  x  =  a  in  Q^t^,  for  which  none  of  the  denominators 
of  the  It's  vanish,  3/  has  at  most  n  values. 

27tus  y  is  at  most  an  n-valued  function.  Its  domain  of  definition 
embraces  all  points  of  9fJ„  except  the  poles  of  the  coefficients  H,  and 
those  points  for  which  i)  has  no  real  root. 

Functions  of  Several  Variables  in  General 

230.  We  can  give  now  the  definition  of  a  function  in  n  vari- 
ables. Let  x  =  (^x^,  x^,  •••  Xn)  range  over  the  points  of  a  certain 
domain  i),  viz.  over  9?„  or  a  part  of  it.  Let  a  law  be  given  which 
assigns  to  y  one  or  more  values  for  each  point  of  D.  We  say  1/  is 
a  function  of  x^,  x^,  •■•  x^,  and  write 

^=/(^i"-^n>  or  y  =  </)(a;i"-a;„),  etc. 

When  no  ambiguity  can  arise,  we  may  even  write 

y=/(a^).  y  =  4><i^~)^  etc., 

where  x  stands  for  the  n  variables  x^--- x^. 

The  meaning  of  the  terms  of  189,  viz.  one-valued  and  many- 
valued.,  independent  variables  or  argument,  dependent  variable, 
domain  of  definition  of  the  function,  when  applied  to  several  vari- 
ables, needs  no  explanation. 

231.  We  explain  now  the  graphical  representation  of  functions 
of  several  variables. 

We  take  w  +  1  axes,  one  for  each  of  the  variables  y,  x^,  x^,  •••  x^. 

.0  ,«! 


y- 


The  representation  of  a  point  x^  =  a^  •••  x^  —  a^,  is  a  complex  of 
n  points  a^ •••«„,  as  in  the  figure.  The  value  of  y,  say  y=b,  is 
represented  by  the  point  b  on  the  ^/-axis.  This  representation, 
although  unsatisfactory  in  some  respects,  is  still  often  useful. 


144    ELEMENTARY   FUNCTIONS.     NOTION    OF   A   FUNCTION 


232.    When  n  =  2,  we  have  two  other  modes  of  representation, 
Let  the  function  be 

We  take  three  axes,  x-^,  x^,  ?/,  as  in  ana- 
lytic geometry,  of  three  dimensions.  To  the 
set  of  values  of  the  independent  variables 


corresponds  the  point  a  =  (a^,  a^)^  whose 
coordinates  are  a^,  a^-  The  value  5  of  ?/  at  this  point  we  lay  off 
on  the  ordinate  through  a.  As  x  runs  over  its  domain  i>,  ?/  will 
ordinarily  trace  out  a  surface  in  D^g. 

233.    The  other  mode  of  representation  is  by  means  of  a  plane 
and  an  axis. 

The  domain  of  the  independent 
variables  we  represent  by  points  in 
the  x-j^x^  plane,  wdiile  ^  is  represented 
by  points  laid  off  on  a  separate  axis, 
as  in  the  figure. 


234.    When   n  =  3,   we  may  employ 
the  following  representation. 
Let 


i/ = /(x^x^x^) 

be  defined  over  a  domain  D. 

To  represent  D,  we  take  three  rec- 
tangular axes. 

To  the  set  of  values 


',a 


Xn   Ctn 


corresponds  the  point  a,  whose  coordinates  are  a^,  a^, 
values  of  y  we  lay  off  on  a  separate  axis,  as  in  the  figure. 


The 


235.  From  the  elementary  functions  of  one  variable  we  can 
build  an  infinity  of  functions  of  several  variables.  We  give  some 
examples  which  illustrate  the  various  domains  of  definition  that  a 
function  of  several  variables  may  have.     We  shall  take  w  =  2. 


COMPOSITE   FUNCTIONS 


145 


Ex.  1. 


For  points  within  the  ellipse  E^  whose 
equation  is  22 

^'-1  =  0, 


x"      y 


the  argument  of  2  is  negative.  For  points  on  E  the  argument  is  0. 
As  the  logarithmic  function  is  defined  only  for  positive  values  of 
the  argument,  the  domain  of  definition  i),  of  3,  is  the  region 
shaded  in  the  figure.     Its  edge,  or  E^  does  not  belong  to  B. 


Cs—  1  )  =  log-  uv. 


236.  Ex.  2. 

Since  log  uv  is  not  defined,  unless  mu  >  0,  m  and  v  must  be  both 
positive,  or  both  negative.  The  domain  of  definition  i>,  of  2,  is 
thus  the  region  shaded  in  the  figure.  Since 
Mz;  =  0  on  the  edge  of  i>,  these  points  do  not 
belong  to  D. 

237.  Ex.  3. 

z  =  tan  \  TTxy. 
Since  tan  u  is  not  defined  when 

u  =  — h  niTT, 

2 

m  =  0,  ±1,  ±2,  ... 

we  see  the  domain  of  definition 

of   z   includes   all   the   points    of 

the   xy   plane,    except    a    family 

of  hyperbolas  ^  ^ 

"^  ^  xy  =  lm  +  1. 

Composite  Functions 

238.  1.   An  extremely  useful  notion  in  many  investigations  is 

that  of  -A  function  of  functions^  or  composite  functions.      Let 


146    ELEMENTARY   FUNCTIONS.     NOTION   OF   A   FUNCTION 
be  defined  over  a  domain  X  in  w-dimensional  space  9?„.     Let 

be  a  point  in  an  w-dimensional  space  ^^. 

While  X  runs  over  JT,  let  u  run  over  a  domain  U.     Let 

y  =  4>(u^'--Um^  (1 

be  defined  over  JJ.  Then  y  is  defined  for  every  point  x  in  X. 
We  may,  therefore,  consider  y  as  a  function  of  the  x's  through  the 
w's.     We  say  ?/  is  a  function  of  functions,  or  a  composite  function. 

2.  When  speaking  of  composite  functions,  we  shall  always  sup- 
pose, even  without  further  mention,  that  the  domain  of  definition 
of  1)  is  at  least  as  great  as  U. 

3.  When  x  ranges  over  X,  u,  as  we  said,  runs  over  the  domain 
U.     It  is  convenient  for  brevity  to  call  U  the  image  of  X. 

Example.  ^ 

Ml  =  xiX2-i  M2  =  sec  Xi,  Us  =  e^i. 

y  =  log  Ml  +  tan  — • 
Here  mi,  U2,  ms  are  defined  for  all  the  points  of  9?2,  for  which 
a;i  T^  0  or  ^  +  rtnr,        m  =  0,   ±1,   ±2,  — 
while  y  is  defined  for  all  the  points  of  9?3,  for  which 

Ml  ^  0,  and  —=^-  +  mr.        n  =  ±  1,   ±  2,  .- 

239.  The  notion  of  a  composite  function  is  sometimes  useful  in 
transforming  a  function  as  follows.     Let 

i/  =  F(x^-"xJ. 

The  variable  x  may  enter  F  in  certain  combinations,  so  that  if 
we  set 

y  goes  over  into  ^  ^  . 


Example. 


Let 
then 


M  =  ^, 
X2 

y  =  aM2  -  l^t-^  +  log  M  =  G(u). 


LIMITED   FUNCTIONS  147 

Limited  Functions 

240.  Let/(a;j  •••rr^)  be  detined  over  a  domain  i).    If  there  exists 
a  positive  number  M,  such  that 

\f\<M, 

for  every  point  of  D,  f  is  said  to  be  finite  or  limited  in  J) ;  other- 
wise /  is  unlimited  in  D. 

Ex.  1. 

f(x)  =  sin  X. 

Since 

I  sin  a;  I  <1,  as  arbitrary, 

sin  a;  is  a  limited  function  for  any  domain. 

Ex.  2. 

is  limited  in  any  domain  (—  G,  G),  where  G  is  some  fixed  positive  number. 
It  is  unlimited  in  the  domain  (0,  +oo),  for  example. 

Ex.  3.  ^,  N      1 

is  defined  for  every  x  ^  0.  * 

It  is  limited  in  any  domain  as  (a,  oo),  if  a  <  0. 
It  is  unlimited  in  (0*,  1),  for  example. 

241.  Letf^x^  •••  a:„),  g{x^  •••  x^)  he  limited  functions  in  a  domain 
D.     Then  j.  n 

f±9^      fg 

are  limited  in  D. 

If  \9\>a>0 

in  D,  then  ^ 

9 

is  limited  in  D. 

Since  /,  g  are  limited  in  D,  let 

\n\9\<^- 
^^"'^  i/±^i<i/i+i^i<2i^. 

Hence/±^  is  limited  in  D. 

^^'"^  \f9\  =  \f\-\9\<MK 


Hence  fg  is  limited  in  D. 
Finally, 


/.     .    . 
Hence  —  is  limited  in  D. 
9 


~\9\      ^' 


CHAPTER   V 
FIRST  NOTIONS  CONCERNING  POINT  AGGREGATES 

Preliminary  Definitions 

242.  In  elementary  mathematics,  the  functions  employed  are 
usually  defined  by  simple  analytic  expressions.  Their  nature  is 
simple,  and  their  domains  of  definition  receive  little  attention.  In 
the  theory  of  functions  we  take  a  higher  standpoint,  and  consider 
functions  defined  by  any  law,  as  explained  in  189  and*  230.  Such 
functions  are  not  tied  down  to  an  analytic  expression ;  indeed, 
we  may  not  know  how  to  form  their  analytic  expressions. 

From  this  point  of  view,  the  domain  D  over  which  the  function 
is  defined  or  spread  out  is  often  of  great  importance.  Frequently 
we  choose  first  the  domain  _Z),  and  then  define  a  function  for  the 
points  of  B. 

The  domain  of  definition  of  a  function  of  n  variables  may  be  any 
set  or  aggregate  of  points  in  9?,i-  We  wish  to  treat  now  the  most 
elementary  properties  of  such  aggregates  which  we  call  point 
aggregates. 

243.  1.   Two  point  aggregates  A^  B  are  equals  when  every  point 

of  A  lies  in  ^,  and  every  point  of  B  lies  in  A.     In  this  case,  we 

write  .       ^ 

A  =  B. 

2.  If  every  point  of  B  lies  in  A  but  not  every  point  of  ^  lies  in 
B,  we  say  5  is  a  partial  or  sub-aggregate  of  A,  and  w;rite 

^  >  J5  or  ^  <  J.. 

3.  If  A  does  not  exist,  i.e.  if  it  contains  no  points,  we  write 

^  =  0. 
148 


PRELtMlNARY  DEFINITIONS  149 

As  the  symbol  0  also  stands  for  the  origin,  we  shall  write,  in 

case  of  ambiguity, 

A  =  (0), 

when  we  wish  to  indicate  that  A  consists  of  the  origin  alone. 
The  fact  that  A  contains  at  least  one  point,  we  indicate  by 

A>0. 

4.  Let  A,  B  be  two  point  aggregates  having  no  point  in  com- 
mon.    The  aggregate  formed  by  their  reunion  is  called  their  swm, 

and  is  denoted  by  ,       ^ 

-^  A+  B. 

5.  If  B  is  a  partial  aggregate  of  A,  the  aggregate  formed  by 

removing  all  the  points  of  B  from  A  is  called  the  difference  of  A, 

B.  and  is  denoted  by  ,       ^ 

■^  A  —  B. 

It  is  also  called  the  complement  of  B. 

6.  If  a  or  X,  for  example,  are  general  symbols  for  the  points  oi 
an  aggregate,  we  can  represent  the  aggregate  by 

\a\   or   ]x\. 
Thus,  if 

■^  —    1'    2'    31 

we  can  write 

Or  if  A  =  Oi,  02,  as,'" 

we  can  write  A  —  {«„}. 

244.    Definitions  of  configurations  in  n-way  space.     Cf.   227,  2. 
1.  Let  a  =  (aj  •••«„),  h  =  (h^---h^')  be  two  points  of  Qf^n.     We  say 

is  the  distance  between  a,  h  ;  we  denote  it  by 

Dist  (a,  5)  or  a,  h. 

2.  The  points  x  satisfying 

2^1  -  «i  =  >-(*i  -  «i)      •••      a:,/,  -  a^  =  X(5„  -  a„)  (2 

lie  on  a  rigJit  line  i,  viz.  the  line  determined  by  the  two  points 
a,  h.     Here  X  runs  over  all  the  numbers  of  9?. 


150      FIRST   NOTIONS   CONCERNING   POINT   AGGREGATES 

When  X  =  0,  x=  a\  when  X  =  1,  x  —  h.  Points  x^  for  which 
0<\<1,  form  a  segment  or  interval  (a,  6)  of  i.  Such  points  are 
said  to  lie  between  a,  h. 

An  aggregate  lying  on  a  right  line  is  called  rectilinear. 

3.  If  three  points  a,  5,  c  lie  on  a  right  line,  we  have  from  2) 

that 

_i '  =  f ^;         t,  «  =  1,  2,  •••  n.  C^ 

0,  -  «,      o«  -  «« 

and  conversely,  if  3)  holds,  a,  5,  c  lie  on  a  right  line. 

4.  Let  a,  5  be  two  points  on  the  line  i,  and 
r=  Dist  (a,  5). 


Then 


«i  -  *i  .        «„  —  ^. 

Aj  =  •  •  •     A„  —  


r  r 

are  the  direction  cosines  of  the  line  L.     Obviously, 

5.  The  points  x  defined  by 

(a;^_a,)2+...+(a;„-a„)2=r2,         r>0. 

lie  ow  a  sphere  8  whose  center  is  a  and  whose  radius  is  r. 
The  equation  of  S  may  also  be  written 

Dist  (a,  x^  =  r. 

The  points  a;,  such  that 

Dist  (a,  a;)<r 
lie  within  tS.     If 

Dist  (a,  x~)  >  r, 

a:  lies  without  S.     If  a;  lies  on  or  within  tS,  it  lies  «w  /S^. 

6.  The  points  x,  such  that 

form  a  CM6e,  with  center  a  and  edge  e. 


PRELIMINARY  DEFINITIONS  151 

7.  The  points  a;,  such  that 

form  a  Tectsnigular  parallelopiped  or  cell  whose  edges  are  of  length 


8.  The  cube 


kl-«li<~     •••     kn-««|<-^ 


is  called  the  inscribed  cube  0  of  the  sphere  S,  of  radius  r,  and 
center  a. 

Every  point  a;  of  C  is  in  S.     For, 

9.  Let  the  cube  (7  be  given  by 

The  points 

f  1  =  «!  ±  I  o-    •  •  •    Vn  =  a„  ±  I  fl- 
are called  the  vertices. 

is  one  vertex, 

v'  =  C-v^  +  2a^, v«  +  2a„) 

is  called  the  opposite  vertex. 

The  line  joining  a  pair  of  opposite  vertices  evidently  passes 
through  the  center  of  C.     It  is  called  a  diagonal. 

The  length  of  a  diagonal  is 


Vo-2  + 1-  0-2  =  cr  Vw. 

10.  The  distance  between  two  points  a,  5  in  C  is  greatest  when 
they  are  opposite  vertices.  For,  each  terra  (a^  —  b^  in  1)  has  then 
its  greatest  value,  viz.  a^. 

11.  If  ^j,  ^2'  "•  ^m  ^'I'e  the  lengths  of  the  edges  of  the  parallelo- 
piped  in  6,  we  say  the  product 

gj  •  ^2  •  •  •  ^rt 

is  its  volume. 

In  case  the  parallelopiped  is  a  cube  of  edge  o-,  its  volume  is  o-". 


152     FIRST   NOTIONS  CONCERNING   POINT  AGGREGATES 

12.  The  points  x  defined  by 

«i.ri  + 1-  a,,x„  +  (f  =  0  (4 

lie  in  a  plane.     The  two  planes  1)  and 

a^x-^  +  •  •  •  +  (in^n  +  e  =  0 
are  parallel. 

245.    Let  a,  5,  c  be  three  points  in  $R„. 

Let 

A  =  Dist  (b,  c),    B  =  Dist  (a,  c),    0=  Dist  (5,  a). 

When  w  =  1,  2,  3,  we  have 

A^B  +  C.  (1 

Here   the  inequality  sign   holds  unless 
A,  B,  C  lie  on  a  right  line  L. 

We  show  now  that  1)  holds  for  every  n.* 
To  this  end,  set 

«i  =  ^i  —  Cli      /3t  =  ftt  —  ^o      7i  =  ^i  —  «t, 

where  «^,  6,  (?^ ;    t=l,  2,   •••  w,  are   the    coordinates  of   a,   5,   c. 
J'hen 

Now,  A^  =  a^^+---  +  «„2  ^  ;s«^2  . 

C2=  7^2+... +7,2^^7.2.  (4 

From  2),  we  have  also 

^2  =  2(^^  +  7^)2  =  v^^2  +  ^7,2  +  2  E/3,7,.  (5 

Thus  to  prove  1),  we  liave  to  show  that 
A^<B^+  (72-f  2i?(7, 
or,  using  3),  4),  5),  that 


2A2  +  E7.2  +  22^.7,<2/32+S7.2  +  2V(/3i2+---+A.0(7i'+---  +  7«')- 

*  If  the  reader  finds  the  demonstration  difficult,  let  him  go  through  it,  taking 
n  =  2  or  3.  We  recommend  this  in  the  case  of  any  demonstration  which  the  reader 
finds  too  hard  for  general  n. 


PRELIxMINARY   DEFINITIONS  153 

This  shows  that  it  will  suffice  to  prove  that 

To  do  this,  we  start  from  the  inequality 

(A7«-/3.7.)'>0.  (7 

By  244,  3,  the  inequality  sign  holds  for  at  least  one  pair  of 
indices  t,  a:,  unless  a,  6,  c  lie  on  a  right  line. 
From  7),  we  have 

Let  us  form  all  the  relations  of  this  type,  letting  i,  k  run  over 
the  indices  1,  2,  -••,  w,  and  keeping  L=f^  k. 
If  we  add  these,  we  get 

2^,V>2  2/3A7c7.-         ^^>C'  (9 

On  the  other  hand, 

2yS,2  .  S7,2  =  (^^2  +  . . .  +  ^^^2)  (^^2  +  . . .  +  ^^2)  ^  ^^2^^  +  2^,2^,2. 

Hence,  by  9), 

2A'  •  27.^  ^  2^^2^^2  +  2  ^/3j3^ry^y^.  (10 

But  '^-^ 

(SyS.x)'  =  C/3i7i  +  •  •  •  +  /3„7  J'  =  2/9,27;^  +  2  2^^^,7,7,. 

This  in  10)  gives  6). 

246.  A  point  x  for  which  Dist  (a,  a;)  is  small,  is  said  to  be  near 
a.  What  is  to  be  considered  as  small,  depends  on  the  problem  in 
hand. 

The  points  x,  such  that 

Dist  (a,  x')  <  p,         p  >0. 

form  an  aggregate  called  the  domain  of  the  point  a,  of  norm  p.     It 

is  denoted  by  t^  ^  -.  -n,  -.  t, 

•^  Dp(a),         D{a),         Dp. 

For  example,  in  9?j,  Dp(^a')  is  the  interval  (a  —  p,  a  +  p')- 
In  9?2'  Dp(^a')  embraces  all  points  in  a  circle  of  radius  p,  and 
center  a  j  in  9^3,  it  embraces  all  points  in  a  sphere  of  radius  p. 


154      FIRST   NOTIONS   CONCERNING   POINT   AGGREGATES 

We  sometimes  wish  to  exclude  the  point  a  from  its  domain. 
When  this  is  done,  the  domain  is  said  to  be  deleted;  we  denote  it 

by 

I)*{a)     or     i)*(a). 

247.  Let  ^  be  a  point  aggregate  in  9?^. 

Let  'p  be  any  point  in  9?„.  We  say  'p  is  an  inner  point  of  A  if 
every  point  in  some  domain  of  p  lies  in  A^  i.e.  if  there  exists  a 
p  >  0  such  that  every  point  of  i^p( jo)  lies  in  A.  The  point  p  is  an 
outer  point  of  A  if  no  point  of  DpCp}  li^s  in  A.,  however  small 
/>  >  0  is  taken.  Finally,  p  is  a,  frontier  point  of  A  if  in  every  Dp(^p^)., 
however  small  p  >  0  is  taken,  there  is  at  least  one  point  of  A  and 
one  point  not  in  A.  Every  point  of  9^„  is  either  an  inner,  an 
outer,  or  a  frontier  point  of  A.  The  frontier  points  of  a  cube  or 
parallelopiped  form  its  surface. 

V-i  Pi  Pi 

248.  Ex.  1.  A=(a,^).  ^    ^ ^ -^ 

Here  any  point  pi,  such  that  «  <i5i  <  /3,  is  an  inner  point.    Any  point  p^-,  such 
that  i)2  >  /3  or  j?3  <  a,  is  an  outer  point. 
The  frontier  points  are  a  and  /3. 

Ex.  2.  A  embraces  the  rational  points  in  (a,  j3).  Here  all  points  p,  such  that 
j9  <  a  or  p  >  ^,  are  outer  points.  The  points  of  A  are  all  frontier  points.  For,  if 
a  be  any  point  of  A,  there  are  irrational  points  in  every  Dp{a),  however  small 
p>  0  is  taken,  by  84. 

In  this  example  A  contains  no  inner  points. 

Ex,  3.    A  embraces  all  the  points  in  9t2,  both  of  whose  coordinates  are  rational. 

Here  every  point  p  of  Siz  is  a  frontier  point.  In  fact,  consider  a  little  circle  G  of 
radius  p  >  0  and  center  p.  Evidently  p  contains  points  in  A  and  points  not  in  A, 
however  small  p  is  taken. 

In  this  example  there  are  no  outer  and  no  inner  points  of  A. 

249.  Let  h  he  an  inner  point  of 

Then 

A  =  A(5) 
lies  within  S  if 

/tJ  +  S  <  0-, 
where 

/3=  Dist  (a,  5). 


PRELIMINARY   DEFINITIONS  155 

The  theorem  is  proved  if  we  show  that  the  points  y  of  A  satisfy 

the  relation  t>.-     ^       x 

Dist  (a,  y)  <  a.  (2 

But,  by  245, 

Dist  (a,  y) <  Dist  (a,  V)  +  Dist  (6,  y)=p-\-h. 
Thus,  by  1),  the  relation  2)  is  valid. 

250.    1.   Let  vl  be  a  point  aggregate,  and  je>  any  point  in  9?„. 

The  points  of  A^  lying  in  -Z>p(j9),  form  the  vicinity  of  p^  of  norm  p. 

It  is  denoted  by  rr  ^    ^         tt-^    x 

^  V^p}  or    V(p). 

Thus  D(p^  embraces  all  points  near  ^,  while  F(j9)  includes  only 
points  of  A,  near  p. 

Example.     Let  A  =  1,  I,  ^,  ••• 

Here  Z>p(0)  is  the  interval  (— p,  p),  while  Fp(0)  is  the  set  of  points 

J_     1_     1_ 

?w '   m  +  1 '   OT  +  2 

where  m  is  the  least  integer  such  that  —  <  />• 


m 


The  point  p  may  or  may  not  lie  in  F^jo).  We  sometimes  wish 
expressly  to  exclude  it.  When  this  is  done,  the  resulting  aggre- 
gate is  the  deleted  vicinity  of  p;  it  is  denoted  by 

V*{p)  or  r*(^). 

251.  When  treating  functions  of  a  single  variable  a:,  we  have 
often  to  consider  the  behavior  of  the  function  on  one  side  of  a 
point  a.  This  leads  us  to  split  the  domain  and  vicinity  of  a  into 
two  parts,  forming  a  right  and  left  hand  domain ;  a  right  and  left 
hand  vicinity  of  a. 

The  right  hand  domain  and  vicinity  we  denote  respectively  by 

BI)(a~),   RVCa}. 

The  left  hand  domain  and  vicinity  are  denoted  by 

XZ)(a),   LV(a). 

The  point  a  lies  in  both  the  right  and  left  hand  domain.  It  lies 
in  both  the  right  and  left  hand  vicinity  if  a  lies  in  F(a).  It 
should  be  remembered  that  these  terms  refer  only  to  rectilinear 
aggregates. 


156      FIRST   NOTIONS   CONCERNING   POINT   AGGREGATES 

252.  1.  A  point  aggregate  is  said  to  be  finite  when  it  contains 
only  a  finite  number  of  points.     Otherwise  it  is  irifinite. 

A  point  aggregate  A  is  said  to  be  limited  when  all  its  points  lie 
within  a  certain  sphere  or  cube,  having  tlie  origin  as  center. 

This  definition  is  equivalent  to  saying  that  the  coordinates  a^ 
a^-,  •••  dfii  of  every  point  of  Jl,  are  numerically  less  than  some  i)Osi- 
tive  number  M.  It  A  is  not  limited,  it  is  said  to  be  unlimited. 
Obviously :  Every  finite  aggregate  is  limited. 

Ex.  1.  A  =  l,  2,  3,  ... 

is  an  infinite  unlioiited  aggregate. 

Ex.  2.  A  =  1,  -h,  I,   ••• 

is  an  infinite  limited  aggregate. 

Ex.  3.  A  =  points  of  the  interval  («,  /3) 

is  an  infinite  limited  aggregate. 

2.  In  the  case  of  a  rectilinear  aggregate  A,  it  may  happen  that 
the  coordinates  of  all  its  points  x  are  less  than  some  number  M. 
We  say  A  is  limited  to  the  right. 

If  the  coordinates  of  all  the  points  x  are  greater  than  some 
number  AT,  we  say  A  is  limited  to  the  left. 

Ex.  1.  ^  =  10,  9,  ...,  2,  1,  0,   -1,   -2,    -3,   ... 

is  limited  to  the  right. 

Ex.2.  A^-5,   -4,   -3,   -2,   -1,  0,  1,  2,  3,  ... 

is  limited  to  the  left. 

253.  1.  It  is  sometimes  convenient  to  divide  an  interval  into 
equal  subintervals  or  a  square  into  equal  subsquares,  and,  in 
general,  an  wi-dimensional  cube  F  into  equal  subcubes. 

For  m  =  1,  2,  3,  this  needs  no  explanation.  When  m  is  >  3,  the 
matter  is  still  very  simple.     The  cube  a:^      ^^ 

F   is   graphically   represented   by  m     ^^  ^^ 

equal  segments  on  the  x-^  •••  x^  axes.      ^2 ^ — ^  "     ^  '        ' 

We  divide  F  into  cubes  whose  sides     -^^  '         ' 

are  1/wth  those    of    F    by    dividing  ^       '^~ 

each  of  these  segments  into  n  equal    ^•^'       ' 

parts.     One  of  these  subcubes  is  then  represented  by  the  points 

which  fall  in  a  set  of  m  segments  as  a^^^  •  ■  ■  aj„^„^. 


LIMITING   POINTS  157 

2.  Instead  of  a  cube  F  in  9=?^,  we  may  wish  to  divide  the  whole 
of  dt„i  into  cubes.     The  meaning  of  this  is  now  evident. 

3.  Let  A  be  any  point  aggregate  in  di^.  Let  us  divide  dim  into 
cubes  of  side  S.  This  also,  in  general,  divides  A  into  partial  aggre- 
gates. This  division  of  A  into  partial  aggregates  we  shall  call  a 
cubical  division  of  A,  of  7iorm  h. 

4.  If  instead  of  dividing  9^^  into  cubes,  we  had  divided  it  into 
rectangular  parallelopipeds  whose  edges  are  ^  S,  we  shall  say  that 
we  have  effected  a  rectangular  division  of  '^^,  of  norm  h. 

5.  The  partial  aggregates,  into  which  A  falls  after  a  cubical  or 
rectangular  division,  may  also  be  called  cells. 


Limiting  Points 

254.  1.  One  of  the  most  impoitant  notions  connected  with 
point  aggregates  is  that  of  a  limiting  point.  Let  ^  be  a  point 
aggregate  in  9?„j.  Any  point  p  of  $)i,„  is  a  limiting  point  of  J.,  if 
however  small  /3>0  is  taken,  Dp{p^  contains  an  infinity  of  points 
olA. 

If  every  domain  of  p  contains  at  least  one  other  point,  p  is  a  limit- 
ing point  of  A. 

For,  let  a  be  a  point  of  A  different  from  p.  Let  0<p<Dist(p,  a). 
Then,  by  hypothesis,  DpQp^  contains  some  points  a^  of  A,  besides  jo. 
Let  0</9j<  Dist  (^,  a^).  Then  Dp  {p}  contains  some  point  a^  of 
A,  besides  p.  Continuing  in  this  way,  we  see  that  the  infinite 
aggregate  of  distinct  points 


all  lie  in  i>p(jt?). 


^l")    ^2'    '^3' 


2.  The  following  may  also  be  taken  as  definitions  of  a  limiting 
point : 

If  Vp{p^  is  infinite,  hoivever  small  p  is  taken,  p  is  a  limiting  point 
of  A;  or, 

If  I^*(j3)>0,  however  small  P  *'«  taken,  p  is  a  limiting  point 
Of  A. 


158      FIRST  NOTIONS   CONCERNING   POINT   AGGREGATES 

3.  If  JO  is  a  limiting  point  of  A  and  p  itself  lies  in  A,  it  is  called 
a  proper  limiting  point.  If  p  is  not  in  J.,  it  is  called  an  improper 
limiting  point. 

An 3^  point  of  an  aggregate  A  which  is  not  a  limiting  point  is  an 
isolated  point. 

4.  Let  ^  be  a  rectilinear  aggregate,  and  a  one  of  its  limiting 
points.  If  no  point  of  A  falls  in  (a*,  a  +  8)  or  in  (a  — S,  a*),  B>0 
sufficiently  small,  a  is  called  a  unilateral  limiting  point.  Other- 
wise a  is  a  bilateral  limiting  point. 


255.  Ex.  1. 


A  —  ■{      1      1      1 
.^  —  -"^i   25  T'  ¥' 


Here  the  origin  is  a  unilateral  limiting  point  of  A.     As  0  does  not  lie  in  A,  it 
is  an  improper  limiting  point. 

Ex.2.  ^  =  0,  1,  1,  i,  ^,... 

The  origin  is  a  proper  unilateral  limiting  point  of  A. 

Ex.  3.  A  =  totality  of  rational  numbers. 

Every  point  p  in  9t  is  a  bilateral  limiting  point. 

If  ;9  is  a  rational  point,  it  is  a  proper  limiting  point  of  A.     If  p  is  an  irrational 
point,  it  is  an  improper  limiting  point. 


Lirniting  Points  connected  ivith  Certain  Functions 

256.    We  give  now  a  few  examples  of  point  aggregates  which 
come  up  in  the  study  of  certain  functions. 

Let  y  =  sin  -• 


The  domain  of  definition  of  this 
function  embraces  all  points  on  the 
a;-axis  except  x  =  0. 

It  oscillates  between  —  1  and  +  1. 

The  points 

for  which  y  takes  on  a  particular  value,  as  2/  =  0,  form  a  point 
aggregate  whose  limiting  point  is  a;=  0. 

In  any  domain  of    this  point,  y  oscillates  from  +1  to—  1  aii 
infinite  number  of  times. 


LIMITING  POINTS   WITH   CERTAIN   FUNCTIONS  159 

257.    Let 

j^  =  sin -.  (1 

sin- 
When  a;  =  0,  or  when  * 

sin-=0,  (2 

y  is  not  defined,  since  for  these  points,  1)  involves  division  by  0. 
The  points  x  for  which  2)  holds  are 

±i,    ±^,    ±^,...  (3 

This  is  a  point  aggregate  whose  limiting  point  is  a;  =  0. 

As  X  approaches  one  of  the  points  — ,  y  oscillates  with  increas- 

nir  -J 

ing  rapidity.    At  the  same  time  these  points,  — ,  become  infinitely 

dense  as  x  nears  the  origin.     The  domain  of  definition  of  y  is  the 
a;-axis  except  the  origin  and  the  points  3). 

•> 

258.    Let  1 

y  =  sin  - 


1 

sin 


1 

sin- 

X 


This  expression  does  not  define  y,  because  of  division  by  0, 
when 

x  =  0,  (1 


or  when  x  satisfies 


sini=0,  (2 

X 


sin_ij  =  0.  (3 

sin  - 

X 

The  points  x  defined  by  1)  and  2)  are 

^  =  0,    ±1,    ±-L,   ... 
considered  in  257. 


IGO      FIRST   NOTIONS   CONCERNING   POINT   AGGREGATES 

It  is  easy  to  see  that  the  points  x  defined  by  3)  form  an  aggregate 
B  such  that  each  of  the  points  of  vl  is  a  limiting  point  of  B.     In 

fact,  let  X  approach  the  point  ± As  it  does  so,  sin  -  becomes 

^      nir  X 

smaller  and  smaller  ;  hence  — -  becomes  larger  and  larger. 

sin  - 

X  1 

Thus  in  the  domain  of  the  point  ± — ■-, 


sin. 


sm- 


oscillates  infinitely  often  between  —  1,  1,  and  in  particular  3)  is 
satisfied  infinitely  often. 

Thus,  the  domain  of  definition  of  y  includes  all  points  of  the 
a;-axis  except  the  points  A  and  B. 

About  each  point  of  B^  y  oscillates  infinitely  often.  These 
points  of  infinitely  frequent  oscillation,  themselves  cluster  infi- 
nitely thick  about  each  point  of  A ;  while  the  points  A  cluster 
infinitely  dense  about  the  origin.  Let  the  reader  try  to  picture 
to  himself  how  the  graph  of  y  looks  about  the  points 


±  — ,  and  0. 


259.    1.   The  functions  of  257  and  258  are  formed  from  that  of 
256  by  a  process  of  iteration. 

In  fact,  let 


then 


Similarly, 


y=  sin-=  6(^x'): 


.    r    1 


sm 


.    1 

sm- 


sm. 


.      1 

sm- 


.    1 

sm- 

Xj 


=  oiecx^l 


=  e\0ie<ix)']U  etc. 


LIMITING   POINTS   WITH  CERTAIN   FUNCTIONS 


161 


It  is  customary  to  write 

exx-)  for  e[e(x)-], 

6\x)  for  6'[^2(^a;)],  etc. 
2.  As  another  example  of  iteration,  let 
y=  loga;=  d(x). 

6'^(x)  =  log(log  x'), 

6^(x)  =  \ogl\.og(logx')'],  etc. 


Then 


260.    Let 


y=  sin-=  ^(a;). 


We  have  noted  the  domain  of  definition  of 
^(a;),   e\x},  d\x~). 

In  general,  let  A^  be  the  domain  of  definition  of  6"^(x').  Each 
point  of  A,„_i  is  a  limiting  point  of  ^,„,  and  0'"{x^  oscillates  in- 
finitely often  about  each  point  of  A,n. 

261.  To  get  functions  of  two  variables  having  more  compli- 
cated domains  of  definition,  we  may  apply  the  process  of  iteration 
to  the  function  of  237. 

Let  ^(2^3/)  =  tan  ^  irxy. 


We  saw  6  was  not  defined  for  points  on  the 
family  of  hyperbolas 

IT)  a;^=2m-j-l.         w  =  0,  ±1,  ±2,  ••• 

Let  us  consider  the  domain  of  definition  Dg  o^ 

G^^xy)  =  tan  (|-  ir  tan  \  irxy^ . 

Through  any  point  P  of  one  of  these  hyperbolas  H^  pass  an 
arbitrary  right  line  L. 

At  any  point  Q  on  the  line,  such  that 

^(2;^)=  2w-t-l,         ?i=0,  ±1,  ••• 
ffi'  is  not  defined. 

But  the  points  Q  have  P  as  limiting  point. 


162      FIRST   NOTIONS   CONCERNING  POINT   AGGREGATES 

Derivatives  of  Point  Aggregates 

262.  1.  Let  ^  be  a  point  aggregate.  The  limiting  points  of 
A,  if  it  has  any,  form  an  aggregate,  which  is  called  the  first  deriv- 
ative of  A^  and  is  denoted  by  A' . 


CjJL.     J.. 

._3     4     5     6          _\n-\-\\ 
2'    3'    4'    5'             \     11     ] 

5 

A'  =  \. 

Ex.  2. 

._1      3      1      4      1      5      1      6 

2'    2'    3'    3'    4'    4'    5'    5' 

^'  =  0,  1.                              -        ■ 

_  1  1      n  +  l 
1  n'       « 

2.   When  tI  is  a  finite  aggregate. 

A'  =  Q. 

But^ 

may  be  infinite  and  yet  have 

no 

derivative. 

Ex.  3. 

is  such  an 

^  =  1,  2,  3,  4, 
aggregate. 

... 

263.  1.  The  first  derivative  A'  may  have  limiting  points;  their 
aggregate  is  called  the  secorid  derivative  of  A.     It  is  denoted  by  A" . 

A"  may  have  limiting  points;  these  give  rise  to  the  third  deriv- 
ative A'",  etc. 

Ex.  1.  fill 

^  =  J— +  -1;       m,  n  =  1,  2,  3,  ... 

[m     n ) 

Let  m  be  arbitrary  but  fixed  ;  then  —  is  a  limiting  point  of  A.  For  A  contains 
the  points 

1  +  1      1  +  1      1  +  1     1  +  1     ... 
m  TO     2      m     3     m     4 

whose  limiting  point  is  obviously  — . 
Thus,  ^ 

A'  =  |o,  1|;        TO  =  1,  2,  3,  ... 

while 

A"  =  (0),  (1 

and 

A'"  =  0.  (2 

As  explained  in  243,  3,  the  equation  1)  means  that  A"  consists  only  of  the  origin, 
while  2)  indicates  that  A'"  has  no  points  at  all. 


DERIVATIVES   OF   POINT   AGGREGATES  168 

Ex.  2.  A  =  rational  points  in  an  interval  7; 

A'  =  I.    See  255,  Ex,  3. 
A"  =  /,  A'"  =  I,   ... 
Thus  A  has  derivatives  of  every  order,  each  being  I. 

2.  If  A,  A',  "•  ^('«>  >0,  while  J.c^+i)  =  0,  A  is  of  order  m. 

264.  Every  limited  infinite  point  aggregate  has  at  least  one  limit- 
ing point. 

1.  For  simplicity  let  us  consider  first  the  case  that  the  aggre- 
gate A  lies  in  the  interval  /=  (a,  5).  We  divide  I  into  halves. 
One  of  these  halves,  call  it  J^  contains  an  infinity  of  points  of  A. 
Divide  I^  in  halves.  One  of  these  halves  must  contain  an  intinity 
of  points  of  A.  In  this  way  we  may  continue  bisecting  each  suc- 
cessive interval,  without  end.  We  get  thus  an  infinite  sequence 
of  intervals  ^ 

each  lying  in  the  preceding,  whose  lengths  converge  to  0. 

By  127,  2,  the  sequence  1)  determines  a  point  a.  This  point  a 
lies  in  every  interval  of  1).  Since  each  D(ol)  contains  some  /„, 
it  contains  an  infinity  of  points  of  A.  Hence  «  is  a  limiting  point 
of  ^. 

2.  The  extension  of  this  demonstration  to  9?„  is  now  readily 
made.     Since  A  is  limited,  it  lies  in  a  certain  parallelopiped  P, 

by  252.     We  divide  now  P  into  two  parts 

«!  <  2^1  <  K^i  -  a^),    ag  <  2^2  <  ^2  "••  «„  <  a:„  <  J„,  (2 

l(5i-ai)<2;i<  5j,    a^<X2<h^  •••  a„<x^<b^. 

In  one  of  these  there  must  lie  an  infinity  of  points  of  A.  To 
fix  the  ideas  suppose  it  is  the  parallelopiped  2)  which  we  call  Q^ 

Q^  differs  from  P  only  in  having  one  coordinate,  viz.  a:^,  restricted 
to  an  interval  half  as  big  as  the  original. 

We  now  divide  Q^  into  two  parts, 

«1  <  2^1  <  K^l  -  "l)'     «2  ^  ^2  ^  K^2  -  ^2)'     «3^^3^^3''"'      (^ 

and 

«i  <  2:1  <  I  (Jj  -  rtj),    i  (62  -  ^2)  <  ^2  ^  h^    ^3  ^  2:3  <  b^  ••• 


164     FIRST  NOTIONS  CONCERNING  POINT  AGGREGATES 

In  one  of  these,  say  it  is  the  parallelopiped  Q^  defined  by  3),  an 
infinity  of  points  of  A  must  lie.  Q^  differs  from  P  in  having  now 
two  of  its  coordinates  restricted  to  intervals  only  half  as  large  as 
the  original  ones. 

We  may  continue  in  this  way  restricting  the  remaining  coor- 
dinates x^  ••'  x^,  to  intervals  half  as  big  as  the  original  ones. 
We  get  parallelopipeds  ^g,  Q^  •••  ^„,  in  each  of  which  lie  an 
infinity  of  points  of  A. 

Let  us  set  P^  =  Q^.  This  parallelopiped  lies  in  P  and  has  each 
edge  just  half  as  big  as  the  corresponding  edge  of  P. 

We  may  now  subdivide  P^  just  as  we  did  P.  After  n  bisections 
we  get  a  parallelopiped  P^^^  which  lies  in  P-^,  which  contains  an 
infinity  of  points  of  A,  and  whose  edges  are  one  half  as  big  as 
those  of  Pj. 

Continuing  this  process  indefinitely,  we  get  a  sequence  of  paral- 
lelopipeds  p    p     P 

which  determines  a  point  «  =  Qct^a^  •■•  a„).    This  point  is  evidently 
a  limiting  point  of  A  by  the  same  reasoning  as  employed  in  1. 

265.  If  A  is  a  limited  aggregate  of  the  nth  order^  ^'"'  is  finite. 
For  if  A^'^'^  were  infinite,  being  limited,  it  must  have  at  least  one 

limiting  point,   by  264.     Then  ^<"+i>  >  0,  which  contradicts  the 
hypothesis. 

266.  Let  A  he  any  point  aggregate.  Then  A"<A',  i.e.  all  the 
limiting  poiiits  of  A'  are  proper. 

1.   For  simplicity,  let  us  first  consider  a  rectilinear  aggregate. 


'Let  p  be  any  point  of  A" ;  we  lay  it  off  on  tlie  A,  A'  axes  also, 
as  in  the  figure. 


DERIVATIVES   OF   POINT   AGGREGATES  165 

To  show  that  'p  lies  in  ^',  we  have  to  show  it  is  a  limiting  point 
of  A,  i.e.  in  any  little  interval  JT  about  J9  there  lie  an  infinity  of 
points  of  A.  Let  /  be  any  little  interval  about  j9  on  tlie  A!  axis. 
As  jo  is  a  limiting  point  of  ^',  /contains  an  infinity  of  points  of  ^ . 
Let  q  be  one  of  these.  Let  us  lay  q  off  on  the  A  axis.  Since  g  is  a 
limiting  point  of  A,  any  little  interval  as  J  contains  an  infinity  of 
points  of  A.  Now,  however  small  K  is  taken,  there  exist  inter- 
vals J  lying  within  K  which  contain  an  infinity  of  points  of  A. 
Hence  K  contains  an  infinity  of  points  of  ^,  and  ^  is  a  limiting 
point  of  A.     Thus  'p  lies  in  A! . 

2.  The  extension  of  this  demonstration  to  9?„j  is  obvious.  To 
show  that  f  lies  in  ^',  we  have  to  show  that  I>J^'p)  contains  an 
infinity  of  points  of  A,  however  small  e  is  taken.  To  this  end,  let 
/3  <  e.  Let  5-  be  a  point  of  A!  in  I)p{p).  Then  D^iq^)  contains  an 
infinity  of  points  of  J.,  however  small  a  is.     But  if 

/3  +  o-  <  e, 

Da(q^  lies  in  i>^(jt?),  by  249.     Hence  B^Qp)  contains  an  infinity 
of  points  of  A. 

3.  We  have  just  shown  that  A"  lies  in  A' .  It  is,  however,  not 
necessary  that  A'  lies  in  A. 

Thus,  if  J.  =  1,  -|,  i,  ••.,  A'  =  (0),  and  this  does  not  lie  in  A. 

267.  Extreme  values  of  a  domain.  1.  Let  the  variable  x  range 
over  a  rectilinear  domain  D  which  is  limited  to  the  right. 

We  form  a  partition  (^,  B^  as  follows :  in  A  we  put  all  num- 
bers of  9?  which  are  ^  any  number  in  D  \  \\\  B  we  put  all  numbers 
of  9^  which  are  >  any  number  of  D. 

Let  this  partition  be  generated  by  fi  [130]. 

We  call  fi  the  maximum  of  x  or  of  i),  and  write 

yu,  =  Max  X  =  Max  D. 

The  fact  that  a  domain  E  is  not  limited  to  the  right  may  be 

denoted  by  ,,  ht       tt 

^  Max  X  =  Max  ^  =  +  00, 

where  x  ranges  over  E. 

2.  Let  D  be  limited  to  the  left.  We  form  a  partition  (tI,  B^ 
by  putting  in  A  all  numbers  of  9?  which  are  <  any  number  of  -Z>, 


166      FIRST   NOTIONS   C0NCI:RNING  POINT   AGGREGATES 

and  in  B  all  numbers  which  are  ^  any  number  of  L.     If  the  num- 
ber X  generates  this  partition,  we  call  \  the  minimum  of  x  ov  of  i), 

and  write  ^       at-  tit-     -rv 

\  =  Mm  X  =  Mm  JD. 

The  fact  that  a  domain  E  is  not  limited  to  the  left  may  be 

denoted  by  ,,.  t.,.     ^ 

•^  Mm  X  =  Mm  i>  =  -  oo, 

where  a;  ranges  over  ^. 

Ex.  1,  D=(a,  b),  a<b. 

Min  x  =  a.        Max  x  =  b. 
We  note  that  as  takes  on  both  its  minimum  and  maximum  values  in  D. 
Ex.2.  i)=(0,  1,  ^,  I,  1,  f,  1,  i,  ...). 

Min  a;  =  0.        Max  x  =  1. 
Here  x  takes  on  both  its  maximum  and  minimum  values. 
Ex.  3.  D=(a*,  b*). 

Min  x=  a.        Max  x  =  b. 

Ex.4.  I)  =  (h  h  h  h  h  h  -)■ 

Minl>  =  0.         MaxZ)  =  l. 
In  Exs.  3,  4,  X  takes  on  neither  its  minimum  nor  its  maximum  values. 

268.  1.  The  maximum  and  minimum  values  of  x  are  called  its 
extreme  values  or  extremes. 

Let  e  be  an  extreme  of  D.  If  the  point  e  is  an  isolated  point 
of  D,  e  is  called  an  isolated  extreme,  otherwise  e  is  a  non-isolated 
extreme. 

Evidently  an  isolated  extreme  of  D  lies  in  D. 

2.  When,  however,  e  is  a  non-isolated  extreme,  it  may  or  may 
not  lie  in  D.     In  this  case  we  have  the  theorem : 

If  e  he  a  finite  non-isolated  extreme  of  i),  it  is  an  extreme  of  D', 
the  first  derivative  of  D. 

To  fix  the  ideas,  let  n.*-      -r. 

e  =  Max  B. 

Since  e  is  not  isolated,  it  is  a  limiting  point  of  D,  and  hence 
lies  in  D' . 

Since  no  a;  of  D  is  >  e,  no  x  of  D'  is  >  e.     Hence 

g=Maxi)'. 


VARIOUS   CLASSES   OF   POINT   AGGREGATES  167 

3.  We  have  obviously  the  following  : 

Let  every  x  of  D  he  ^fi,  while  for  each  e  >  0  there  exists  in  D  an 

x>a  —  €.     Then  , ,       ^ 

fjL  =  Max  2>. 

A  similar  theorem  holds  for  a  minimum. 

4.  Let  9)?  be  such  that 

Mina;<9W<Maxa;; 
we  call  W  a  mean  value  of  x,  or  a  mean  of  x,  and  write 

W  =  Mean  x. 

269.  1.  Let  e  be  an  extreme,  finite  or  infinite,  of  the  function 
f(x-^ '  •  •  x^^  with  respect  to  a  limited  domain  D.  Then  there  exists  a 
point  a,  not  necessarily  in  D,  such  that  e  is  the  extreme  of  f  in  any 
vicinity  of  a,  however  small. 

The  demonstration  is  precisely  analogous  to  that  given  in  264. 
2.  Lf  D  contains  its  limiting  points,  the  point  a  lies  in  D. 

Various  Classes  of  Point  Aggregates 

270.  Each  point  p  of  an  aggregate  A  is  either  an  isolated  point 

or  a  limiting  point  of  A.     Let  us  call  Ai  the  aggregate  of  the 

former  points  and  A;,  the  aggregate  of  the  latter  points. 

Then  .        .         . 

^  =  .4,  +  Ak. 

If  A),  =  0,  then  A  =  A„  and  A  is  an  isolated  aggregate. 

If  A,=  0,  then  A=A^,  and  A  is  dense. 

A  may  contain  all  its  limiting  points ;  it  is  then  complete. 

If  A  is  dense  and  complete,  it  is  perfect.  It  then  contains  all 
its  limiting  points,  and  every  point  of  ^  is  a  limiting  point. 

A  point  aggregate  such  that  each  of  its  points  is  an  inner  point 
is  called  a  region.     Cf.  247. 

It  is  sometimes  convenient  to  consider  the  aggregate  formed  of 
a  region  and  its  frontier  points.  Such  an  aggregate  is  called  a 
complete  region. 

For  example,  the  interior  of  a  circle  forms  a  region.  If  we  add 
its  circumference  we  get  a  complete  region. 


168      FIRST   NOTIONS   CONCERNING   POINT   AGGREGATES 

271.  Ex.  1.  A  =  l,  1,  i,  i,  •.. 
is  an  isolated  aggregate. 

Ex.  2.  ^  =  0,  1,  1,  i,  ... 

is  a  complete  aggregate. 

Ex.  3.   A  =  the  rational  numbers  in  a  certain  interval. 
A  is  dense,  but  7iot  complete. 

Ex.  4.  A  =  tlie  interval  (a,  6). 
A  is  perfect. 

Ex.  5,   ^  =  the  interval  (a*,  &). 
J.  is  dense,  but  no^  complete. 

Ex.  6.    A  region  is  dense,  but  not  complete. 
A  completed  region  is  perfect. 

!Ex.  7.    In  the  interval  (0,  1)  remove  the  points  0,  1,  i,  ^,  ••• 
'    The  remaining  points  form  a  region. 

Ex.  8.    In  the  plane  9^2  let 

A  =  ai,   02,    ••• 

be  an  aggregate  having  a  single  limiting  point  a.  Let  us  suppose  that  a  does  not 
lie  in  A.  About  each  a„  let  us  describe  a  circle  C'„  of  radius  so  small  that  no  tv?o 
circles  have  a  point  in  common.  The  aggregate  formed  of  the  points  of  9^2  within 
each  circle  C„  is  a  region  r„. 

The  aggregate  formed  of  all  the  regions  r„  is  also  a  region. 

272.  1.  An  interesting  example  of  a  rectilinear  perfect  aggre- 
gate lying  in  an  interval  %  and  yet  not  embracing  all  the  points 
of  51  is  the  following,  due  to  Cantor. 

Let 

be  expressed  in  the  triadic  system  [144],  restricting,  however,  the 
numbers  a^,  a^,  •••  to  the  values  0,  2,  Then  A  =  \al  is  such  an 
aggregate,  as  we  now  show. 

We  can  get  a  good  idea  of  this  aggregate  as  follows. 


Let  the  interval  (1,  4)  be  of  unit  length.  We  divide  it  into 
three  equal  segments,  (1,  2),  (2,  3),  (3,  4).  The  points  1,  3  are 
points  of  A.  No  point  of  A  falls  within  the  middle  segment 
(2,  3).     We  have  therefore  marked   this  segment   heavy  in  the 


VARIOUS   CLASSES  OF   POINT   AGGREGATES  169 

figure.     We   now  divide    the    segments   (1,  2)   and  (3,  4)  into 
three  equal  segments, 

(1,5),    (5,6),    (6,2),   and    (3,7),    (7,8),    (8,4). 

The  end  points  6,  8  are  points  of  A.  No  point  of  A  falls  within 
the  segments  (5,  6),  (7,  8),  which  are  therefore  marked  heavy. 

In  this  way  we  can  continue  indefinitely  subdividing  the  seg- 
ments within  which  a  point  of  A  falls.  Consider  the  end  points 
of  a  heavy  interval,  say  the  interval  (5,  6).  Its  right  hand  end 
point  is  obviously  a  number  of  A  having  a  finite  representation. 
Its  left  hand  end  point  is  a  point  of  A  whose  representation  is 
infinite. 

To  show  now  that  A  is  perfect,  let  us  begin  by  showing  that 

every  point  a  of  -4  is  a  limiting  point. 

Let 

a  =  •  a^a^-'-a^ 

be  a  point  having  a  finite  representation  [144,  5]. 
Obviously,  a  is  the  limit  of  the  sequence, 


a'  ='  a-^- 

••«.2, 

a"  =  •  a-^' 

■-a,02, 

a"'='a^- 

..a,002, 

Let 


a  =  •  a■^^a^a^' 


be  a  point  whose  representation  is  infinite.     It  is  obviously  the 

limit  of  the  sequence, 

a'  =  •  a^ 


a    =■  a.a. 


l"'2' 


(X  —   *    Cv-tCtniA/n 


Hence  every  point  of  ^  is  a  limiting  point.  On  the  other  hand 
A  contains  all  its  limiting  points. 

For  every  limiting  point  «  of  ^  is  either,  1°,  an  end  point  of 
the  black  intervals,  or,  2°,  not  such  a  point. 


170     FIRST  NOTIONS  CONCERNING  POINT  AGGREGATES 

In  case  1°,  a  is  obviously  a  point  of  A. 

In  case  2°,  if  « r^t  0,  we  can  find  a  monotone  sequence  of  points 
in  A. 

a'  =  •  aj, 

a"  =  •  a^a^, 

a'"  —•  a^a^a^^ 

whose  limit  is  a. 

On  the  other  hand,  the  limit  of  this  sequence  is 

which  is  a  number  of  A.     Hence,  a  is  in  A.  \ 

Should  a  =  0,  this  method  is  inapplicable.  \ 

But  obviously  a  =  0  is  in  J..  \ 


2.  It  is  easy  to  generalize  the  above  example  as  follows.     Let 

a  =  •  a-,an(i<i"' 


f  1  lAJnlj 


be  expressed  in  an  w-adic  system  restricting  the  numbers  a^,  a^^ 
to  a  part  of  the  system, 

0,  1,  2,  •••  w-1; 
for  example, 

0,  2,  4,  6,  ... 


> 


CHAPTER   VI 

LIMITS  OF  FUNCTIONS 

FUNCTIONS   OF   ONE   VARIABLE 

Definitions  and  Elementary  Theorems 

273.  1.  We  extend  now  the  notion  of  limit,  by  defining 
limits  of  functions.  We  begin  by  considering  functions  of  a 
single  variable  x. 

Let  fQc)  be  a  one-valued  function  defined  over  a  domain  D. 
Let 

be  any  sequence  of  points  of  Z),  such  that 

lim  a„  =  a ;         a  finite  or  infinite,       a„  =f=  a. 
If  the  sequence 

/(«l).    /(«2)'    /(«3)     -  (2 

has  a  limit  77,  finite  or  infinite,  always  the  same,  however  the 
sequence  A  be  chosen,  we  say  ?;  is  the  limit  of  f(x)  for  x  =  a  and 
write 

7;  =  lim /(a;), 

or,  more  shortly, 

?7  =  lim/(a;). 

We  also  say /(a:)  approaches  or  converges  to  rj  as  a  limit,  when  x 
approaches  a  as  a  limit.     This  may  be  expressed  by  the  symbol 

fix)  =  V  ' 

2.  If  for  some  sequence  1)  the  limit  of  2)  does  not  exist,  we 
say  the  limit  oif(x)  for  x=  a  does  not  exist. 

171 


172  LIMITS   OF   FUNCTIONS 

3.  Since  the  limit  of  2)  must  be  r}  however  the  sequences  1) 
are  chosen  (provided,  of  course,  they  have  a  as  limit  anda^^a), 
we  have  the  theorem  : 

Let  A  =  \an\t  B  =  \hn\  he  two  sequences  lying  in  D;  let  a„  =  a, 
hn^a. 

If  lim/(a„):#.lim/(5„), 

then  limf(x).,for  x  =  a,  does  not  exist. 

274.  1.  It  is  sometimes  convenient  to  restrict  the  sequences 
A  =  a-^,  a^,  •••  so  that  all  the  jjoints  a„  lie  to  the  right  of  a.  In 
this  case  we  call  r]  a  right  hand  limit  and  write 

7]  =  lim  /(x)  or  t]  =f(_a  +  0)  or  r]  =  R  lim  f(x}  or  tj  =  R  lim  fQx}. 

x=a+0  x=a 

If  we  restrict  the  sequences  A  to  lie  to  the  left  of  a,  we  call  rj  a 
left  hand  limit  and  write 

T]  =  limf(x^   or  t]  =f(a  —  0)   or  tj  =  L  lim /(a;)  or  rj  =  L  lini/(.t;). 

x=a-0  x—a 

Obviously  if 

lim /"(a:;)  =  i],        finite  or  infinite.  (1 

then 

L  lim  f(x)  =  R  lim  fQc)  =  77.  (2 

Conversely.,  if  2)  holds.,  1)  does  also. 

2.   Right  and  left  hand  limits  are  called  unilateral  limits.     If 

we  do  not  care  to  specify  on  which  side  of  a  the  limit  is  taken,  we 

can  denote  it  by 

U\imf(x). 


275.  1.  When  considering  infinite  limits  or  limits  for  a;=  ±qo, 
it  is  often  convenient  to  suppose  the  axes  terminated  to  the  right 
and  left  by  two  ideal  points  +  00,  or  —  00,  respectively.  We  call 
these  the  points  at  infinity. 

We  call  the  interval  ((7,  +  00)  the  domain  of  +  00,  and  denote 

it  by 

i>«(+^)-  (1 

We  call  G-  the  norm  of  -Z)(+  00). 


DEFINITIONS  AND   ELEMENTARY   THEOREMS  173 

Let  J.  be  a  point  aggregate  lying  on  our  axis.  Those  of  its 

points  which  fall  in  1)  we  call  the  vicinity  of  +  oo  for  the  aggre- 
gate A.      We  denote  it  by 

VcX+oo).  (2 

Similar  definitions  hold  for 

Dg(- oo)  and  Fe(-oo).  (3 

2.  When  G-  increases,  the  intervals  ((r,  +  oo)  or  ((r,  —  oo)  are, 
in  a  way,  diminishing.     It  is  convenient,  for  uniformity,  to  say 

that  ^ 

-Z>(;(±oo),    Va(±cc} 

are  arbitrarily  small  when  (r  is  taken  arbitrarily  large,  positively 
or  negatively,  according  to  the  sign  of  oo. 

276.  Corresponding  to  the  two  ideal  points  ±  oo,  Ave  shall  intro- 
duce two  ideal  numbers,  which  we  also  denote  by  ±  oo.  These 
numbers  are  respectively  greater,  positively  or  negatively,  than 
any  number  in  9?.     We  say  they  are  infinite. 

The  system  formed  by  joining  ±  oo  to  the  system  9fJ  we  denote 
by  ^. 

We  shall  perform  no  arithmetical  operations  with  these  ideal 
numbers. 

277.  1.  Most  of  the  theorems  established  in  Chapters  I,  II 
for  sequences  may  be  extended  easily  to  theorems  on  limits  of 
functions. 

For  convenience  of  reference  we  collect  the  following.  The 
reader  should  remember  that  a  theorem  relating  to  limits  for  a 
point  x  =  a  may  be  changed  at  once  into  one  relating  to  a  left  or  a 
right  hand  limit  at  a. 

2.   Let 

lim  /(a;)  =  a,    lim  g(x)  =  /3.  a  finite  or  inf. 

x=a  x=a 

Then 

lim(/±^)=a±yS, 

lim/^=«^. 
See  49,  50,  51,  98. 


174  LIMITS   OF   FUNCTIONS 

3.   In  F'*(a),  a  finite  or  infiyiite^  let 

/(^)<5'(^)<^(^). 
lim/(a:)  =  lim  h(x)  =  \. 

x=a  x=<t 

lim  g(x)  =  X. 


Let 


Then 
See  107. 
4.  Let 


be  finite.     If 


lim  /(a;)  a  finite  or  inf. 

\<f(x)<fM,         in  V*Qa) 


then  X  <  lim  /(re)  <  /x. 

See  106,  1.  """ 


5.  Let 


lim /(^x)  —  a,    lim  g(^x}  =  ±  oo.     « ^wife  or  inf. 


f 
Then  lim  (/  ±  ^)  =  ±  oo,         lim  -  =  0. 


Ifa=^0, 
See  137. 
6.  Let 


lim  fg=±cc. 


lim  /(a:)  ti^  0,    lim  ^(a:)  =0.        a  finite  or  inf. 


If  gQx)  has  one  sign  in  F*(a), 

lim  —  =  ±  00. 
9 
See  137. 

7,  In  V*(^a^,  a  finite  or  infinite,  let 

/(^)>K^)- 
lim  g(x)  =  +  oo, 

lim  f(x')  =  -f  oo. 
See  138.  '^ 


SECOND   DEFINITION   OF   A   LIMIT  175 

8.  In  F*((i)  let  f(x)  he  limited  and  monotone.     Then 

/(«  +  0),        /(a-0) 
exist  and  are  finite. 

See  109. 

Second  Definition  of  a  Limit 

278-    1-  ^  r      ^^  ^  ^   v         •  ^ 

iim  j{x)  =  ?;,         a  finite  or  inf. 

x=a 

there  exists  for  each  e  >  0  a  vicinity  V*(a)  such  that 

\V-Kx')\<e  (1 

in  V*(ci). 

For  let  D  be  the  domain  of  definition  of  f(x).  Let  A  =  ^f  |  be 
the  points  of  i>,  if  any  such  exist,  for  which  1)  is  not  satisfied. 
Let  us  suppose  at  first  that  a  is  finite.     Let 

Min  ||  —  a[  =  /x. 

If  /A  >  .0,  let  0  <  S  <  /i,  then  1)  holds  in  F'5*(a). 
If  ^  =  0,  let  t     t     t 

61"    62'     63' 

be  a  sequence  in  A,  whose  limit  is  a.     Then 

n=co 

and  this  contradicts  the  hypothesis. 

Thus,  when  a  is  finite,  there  exists  always  a  vicinity  V^*(a)  for 
which  1)  holds. 

Suppose  a  =  +  oo .     Let 

Max  I  =  /x. 

If  fi  is  finite,  let  G-  >  fi.     Then  1)  holds  in  F'(;(4-  oo). 
If /x  =  +  Qo?  let  t     t     t 

6l'>    62'    63' 

be  a  sequence  in  A  whose  limit  is  +  x .     Then 

and  this  contradicts  the  hypothesis. 

A  similar  reasoning  applies  when  a=  —  ao. 

2.  We  wish  expressly  to  note  that  in  passing  to  the  limit  x=  a. 
the  variable  x  never  takes  on  the  value  x=  a. 


176  ,         LIMITS   OF   FUNCTIONS 

279.    The  converse  of  the  theorem  278  is  obviously  true,  viz. : 

If  for  each  e  >  0  there  exists  a  vicinity  V^*(a)^  8  >  0,  a  finite  or 
infinite^  such  that 

k-/(a^)|<e 
in  Ffi*(a),  then 

lini  f(x)  =  rj. 


280.    1.   From  278  and  279  we  see  that  we  can  take  the  follow- 
ing as  definitions  of  a  limit : 

The  limit  of  f(x)  for  x=  a  is  rj  when  for  each  e >  0  there  exists  a 

8  >  0,  such  that 

\f(x)-v\<e  (1 

in  r^*(a). 

This  condition  we  shall  express  as  follows : 

e>0,    8>0,    \f(x)-v\<e,    F,*(a).  (2 

Such  a  line  of  symbols  is  to  be  read  as  above. 

2.  The  limit  of  f(x)  for  a;  =  +  oc  is  r],  when  for  each  e  >  0  there 
exists  a  G->0,  such  that  1)  holds  in  F^(-(  +  ao). 

This  condition  we  shall  express  thus : 

e>0,     (7>0,     |/(a;)-7;|<e,     Vo(  +  cc). 

3.  The  limit  of  fix)  for  a;  =  —  oo  is  t],  when  for  each  e  >  0,  there 
exists  a  G<0,  such  that  1)  holds  in  Vg(^—  oo'). 

This  condition  we  shall  express  thus : 

e>0,    a<0,    \f(x)-v\<e,    r^(-oD). 

281.  I.  If 

lim/(a;)  =  +  00,          a  finite  or  infinite. 

x  =  a 

there  exists  for  each  Gr>0  a  vicinity  F'*(a),  such  that 

f(x^>a  (1 

in  F*(a). 

For,  let  A  =  m  be  the  points  of  i>,  if  any  such  exist,  for  which 
1)  does  not  hold. 


SECOND   DEFINITION   OF   A   LIMIT  177 

1°.  Let  a  he  finite.     Let 

Min  [|  —  a|  =  /u.. 

If  /A  >  0,  let  0  <  S  <  ^.     Then  1)  holds  in  F£*(a). 
If  /A  =  0,  let 

be  a  sequence  in  A  whose  limit  is  a.     Then 

and  this  contradicts  the  hypothesis. 

2°.  Let  a=+cc.     Let 

Max  f  =  /i. 

If  fx  is  finite,  let  Gr> /x;  then  1)  holds  in  VqC+cc). 

If  /u.  is  infinite,  let 

?v    ?2'    ••• 

be  a  sequence  in  A  whose  limit  is  +  oo.     Then 

lii"/(l«)  ^  +  Qo  ; 

and  this  contradicts  the  hypothesis. 

A  similar  reasoning  holds  when  a  =  —  ao. 

The  reader  will  observe  that  this  demonstration  is  analogous  to 
that  of  278. 

2.  The  converse  of  1)  is  obviously  true,  viz.: 

If  for  each  (r  >0,  there  exists  a  vicinity  V*(^a).,  a  finite  or  infinite., 

such  that  ^^  ^       ^ 

•fi^}>(^ 
in  V*(^a').,  then 

lim/(2;)  =  +  GO. 

282.    1.   From  281  we  see  that  the  following  definitions  of  limits 

may  be  taken  :  r      ^^  ^        ,  -f 

lim/(2;)  =  +  CO,  II 

x  =  a 

M>0,     S>0,    f(x)>M,     Vs*(ia), 

which  in  full  means  :   if  for  each  M>  0,  large  at  pleasure,   there 
exists  a  8>0,  such  that  f(^x}> Min  V^*(a'). 


178  LIMITS   OF   FUNCTIONS 

2.  lim/(.r)  =  —  oo,  if 

X  =  a 

i.e.  if  for  each  M<0  there  exists  a  S>0,  such  that /(a;)  <  iHf  in 

3.  lim/(2;)  =  +  oo,  if 

ar=+oo 

M>o,    a>o,  f(x-)>M,    r^(+oo). 

4.  lim/(a;)  =  —  GO,  if 

x=+cc 

5.  ]im/(a;)  =  -f  oo,  if 

a;=— 00 

6.  lira /(a;)  =  —  go  ; 

a-  =  —  CO 

i»f<0,      (7<0,    /(a;)<lf,      F^(-oo). 

7.  The  limit,  finite  or  infinite,  of /(a:)  for  a;=  +  oo  or  —  oo  may 
be  represented  by 

respectively. 

283.  By  the  aid  of  the  ideal  points,  with  their  associate  domains 
and  vicinities,  we  may  sum  up  all  the  preceding  six  cases  in  one 
general  statement : 

7}  =  lim/(2;)  a,  rj  finite  or  infinite, 

when,  BQrf)  being  taken  small  at  pleasure,  there  exists  a  vicinity 
V*((i),  such  thatf(x)  lies  in  D(7])  when  x  runs  over  F'*(a). 

See  275,  2. 

The  reader  should  observe  that  the  two  demonstrations  of  278 
and  281  are  perfectly  parallel.  It  is  easy,  by  employing  the  con- 
vention of  275,  2,  to  formulate  the  demonstration  given  in  278  so 
as  to  include  the  cases  treated  in  281,  and  so  make  the  latter 
unnecessary. 


SECOND   DEFINITION   OF   A   LIMIT  179 

284.   In  order  that  lim/(a;)  exists^  a  finite  or  infinite^  it  is  neces- 

sari/  and  sufficient  that  for  each  e>0  there  exists  a  vicinity  ]^*(«), 
such  that 

i/(^i)-/(^2)i<e'  a 

for  any  pair  of  points  Xp  x^  in  F*(a). 
It  is  necessary.     For,  if 

77  =  lim/(a;), 

1=0 

then  for  each  e> 0  there  exists  a  F*(a),  such  that 

h-/(^)l<| 
for  any  x  in  F'*(a).     Let  a;^,  x^  be  two  points  in  F*(a).     Then 

k-/(^l)l<|'      i'?-/(^2)l<|- 

Adding  these  two  inequalities,  we  get  1). 

It  is  sufficient.     For,  let  a^,  a^,  ■-•  be  a  sequence  of  points  in 
V*(^a},  having  a  as  limit.     Then  the  sequence 

/(«i).  /(«2)'  ••• 

is  regular  by  1).     It  therefore  has  a  limit  t}. 

Then 

e>0,     m',     1 77 -/(«„)  I  <  |-        w>w'.  (2 

Let  B  =  b^,  b^,  •••  be  any  sequence  of  points  in  V*(^a^  whose 
limit  is  a.     Then,  by  1), 

|/K)-/(^„)l<f        n>m".  (3 

Adding  2),  3),  we  have 

1 7;  — /(5„)  I  <  e.      n>m,    m>m',m".      (4 

But  since  B  was  an  arbitrary  sequence,  the  relation  4)  states 

that 

7)  =  lim/(a:). 


180 


LIMITS   OF   FUNCTIONS 


Grajjliical  Representation  of  Limits 

285.  The  graphical  representation  of  limits  of  sequences  ex- 
plained in  43,  44,  and  124  may  be  readily  extended  to  limits  of 
functions. 

Let  the  graph  oif(x)  be  referred  to  rectangular  coordinates. 

Let  D  be  the  domain  of /(a:),  and  let 


lim  fQx)  —  I. 


Then  the  condition 


e>0,  S>0,  \f(x')-l\<e,  Fa* (a), 

has  the  following  geometric  interpretation  : 

About  the  line  y  =  I  construct  a  band 
(shaded  in  the  figure)  of  width  2  e,  e  being 
small  at  pleasure.  Then  there  exists,  corre- 
sponding to  this  e,  an  interval  of  extent  2  h 
(marked  heavy  in  the  figure),  such  that  f(x) 
falls  in  the  e-band  for  each  x^a  of  i>  falling  in  the  S-interval. 
In  general,  as  e  is  made  smaller  and  smaller,  S  becomes  smaller 
and  smaller.  But  for  each  e-band,  however  small,  there  corre- 
sponds a  S-interval  of  length  >  0. 


y 

I 

^^B^^^ 

e 

^^^^^^^^ 

~~o 

5 

a 

5 

286.    Let 


lim /(a;)  =  -f  oo. 


Draw  the  line  y  =  M,  where  ilf>  0  is  large 
at  pleasure.  Then  there  exists,  corresponding 
to  this  Jf,  a  S-interval,  marked  heavy  in  the 
figure,  such  that  f(x)  falls  in  the  Jf-band 
(shaded  in  the  figure)  for  each  x=^a  of  D, 
falling  in  the  S-interval. 

As  iHf  is  taken  greater  and  greater,  the  corresponding  S 
becomes,  in  general,  smaller  and  smaller.  But  for  each 
ever  large,  there  corresponds  a  S-interval  of  length  >  0. 


-interval 
M,  how- 


287.    Let 


lim  f(x)  =  I. 


J 


GRAPHICAL    REPRESENTATION   OF   LIMITS  181 

Draw  the  line  y  =  1^  and  construct  an  e-band,  as  in  figure.     For 
each  e  there  exists  a   (r>0,  such  that 
f(_x)  falls  in  the  e-baiid  for  each  x  of 
i>,  falling  in  the  interval  ((r,  +  oo). 

These  examples  will  suffice  to  illus- 
trate  the   graphical    interpretations    of 
limits,  when  f(x)  is  plotted  in  rectangu-     , 
lar  coordinates. 

288.    1.   When  the  graph  of   y=f(x)  is  given  by  means  of 
two  axes,  as  explained   in   191,  the  geometric  interpretation  of 
limits  of  /(a;)  will  be  made  clear  by  the 
tollowing  :  « — \ 1 — I — ' 

Let  lim  fQc)  =  I.  (1  p e   l   e 

x=a  y  '  "■■  I  I 

About  y  =  1  we  mark  off  the  e-interval ;  about  x  =  a  we  mark 
off  the  3-interval. 

Then  1)  requires  that /(a;)  falls  in  the  e-interval  for  each  value 
ot  x=^a  in  I),  falling  in  the  S-interval. 

2.  Let  ..      ..  . 

limf(x)  =  4-  GO. 

x=a 

On  the  2/-axis  we  mark  off  at  pleasure  the  point  M>  0.  Then 
for  each  M  there  exists  a  S-interval,  such  that  /(x)  falls  in  the 
interval  (iHf,  +  oo),  for  each  x=^  a  oi  D  falling  in  the  S-interval. 

lim  f(x)  =  1.         I  finite  or  infinite. 

X  =  a  +  hu,         S  =/=  0,  (1 

^^'"^  lim  fix)  =  I  (2 

For,  while  x  ranges  over  the  domain  D  on  the  a;-axis,  w  ranges 
over  a  domain  A  on  the  w-axis. 

The  two  axes  x  and  u  stand  in  1  to  1  correspondence  by  virtue 
of  1).  To  the  point  a;=a  on  the  a;-axis  corresponds  the  point 
w  =  0  on  the  w-axis. 


182  LIMITS   OF   FUNCTIONS 

T  pf 

f(x)=Aa^-hu)  =  ,^(u).  (3 

Then  if  x  and  u  are  corresponding  points,  /  has  the  same  value 
at  2;  as  0  has  at  u.     To  fix  the  ideas  let  I  be  finite.     From 

e>0,    S>0,     \l-f{x~)\<e,    in  V,*(ia\ 

follows 

^  \l-4>{u)\<e,    in  r,^*(0),  (4 

where  h,=-h. 
h 

But  from  3),  4)  follows  2). 

lim  f(x~)  =  I.         I  finite  or  infinite. 

Let  1 

x  =  — 
u 

Then 

B\imf(x}  =  l, 

and  conversely.  ^  q 

This  follows  at  once,  as  in  289,  +co 

by  observing  that  to  points  in  the  o     -f/o 

shaded  interval  on  the  a;-axis  corre- 
spond points  in  the  shaded  interval  on  the  w-axis. 

2.  Let 


Let 

Then 
and  conversely. 


lim  f(x)  =  1.         I  finite  or  infinite. 

-1 


X= 

u 


Illimf(x')  =  l, 


291.    1.  As  a  result  of  289  and  290,  we  may,  by  the  aid  of  the 
transformations, 

u  =  ax  +  /3,    and   u  =  -■> 
transform  ^ 

lim   into   lim, 

x=a  u=b 

R  lim    into   R  lim,   L  lim,    or  lim. 


i 


GRAPHICAL   REPRESENTATION   OF   LIMITS  183 

Similarly, 

lim   into   Rlim   or    Llim. 

2.   In  particular,  any  limit  x=a  or   ±00  may  be  transformed 
into  one  with  respect  to  a;  =  0. 

292.    Let  u  =  ^(x^,  and 

lim  u  =  h.        a^h  finite  or  infinite.      (1 
Let  y  =  /(w),  and 

lim  y  =  7).  T)  finite  or  infinite.     (2 

Then  if  <\>(x)4-h  in  F*(a), 

lim  y  =  'r).  (3 


To  fix  the  ideas,  suppose  a,  5,  t]  are  finite.  S  a  3 

Then  since  2)  holds, 


I     I     I 


'     '    ■ 


e>0,    cr>0,     |y-77|<e,     r/(6).  "^ 

But,  by  1),  y 

o->0,    S>0,    0<|m-5|<o-,    V^*ia^. 

Hence  while  x  ranges  over  V^ia).,  y  lies  in  D^rf).     Thus 

e>0,    S>0,     |y-7;|<e,    V,*(ia-). 

But  then  3)  holds. 

The  case  that  any  or  all  the  symbols  a,  6,  77  are  infinite  is  per- 
fectly analogous. 

293.    Let  u  =  ^(a;),  and 

lim  u  =  b.         a  finite  or  infinite. 

Let  y=f(u'),  and 

lim  y  =  r}. 

lim  y  =  Ti. 

x=a 

This  follows  as  in  292. 


184 


LIMITS   OF   FUNCTIONS 


294.    1.  Let  y=f{x)   he  a  univariant  function  in  a  unilateral 
vicinity  V*  of  a.     If 

Ulhmj  =  b,  [274,'-] 

then 

UVimx  =  a.  I 

y 
To    fix   the    ideas,  suppose   y  is  increasing 

in  the  left  hand  vicinity  of  a.  

Let  e  >  0  be  arbitrarily  small,  and 


e  x'  a 


a  —  e  <  x'  <  a. 

Let  y'  correspond  to  a;'.     Let  S  >  0  be  such  that 

b-8>y'. 

Then,  while  y  remains  in  LV&*{b'),  x  remains  in  LD^(a). 

2.  Let  y  =/(a;)  he  univariant  in  F'*(a),  where  a  is  a  bilateral 
limiting  point  of  V*.     If 

lim  y  =  b^ 

then 

lim  x  =  a. 

y  =  b 

The  demonstration  is  analogous  to  that  of  1. 

Examples  of  Limits  of  Functions 
295.    1.  lim  sin  a:  =0. 


For,  however  small  e  >  0  is  taken,  there  exists  an  arc  S  >  0  such 
that 

lsinaj|<e.  \x\<h. 


2. 


lim  cos  2;=  1. 


For,  however  small  e>0  is  taken,  there  exists  an  arc  S>0  such 
that 

1  — cos2'<e.  |a;|<S. 


EXAMPLES   OF   LIMITS   OF   FUNCTIONS  185 

296.  1.  lim  sin  x  =  sin  a.  (1 

a:  =  o 

For,  let 

X  ^=  a  -\-  XL.- 
Then 

sin  X  =  sin  (a  +  m)  =  sin  a  cos  u  +  cos  a  sin  w.  (2 

Since 

lim  sin  x  =  lim  sin  (a  +  m), 

x=a  u=0 

and 

lim  sin  w  =  0,  lim  cos  m  =  1, 

«=0  «=0 

equation  2)  gives,  on  passing  to  the  limit,  1),  by  289. 

2.  Similarly, 

lim  cos  X  =  cos  a. 

x=a 

297.  1.  L  lim  tan  x  =  +  oo.  (1 

For,  in  LV*(7r/2'),  tana;>0. 

sin  x 

As  tan  X  = •, 

cos  a; 

and  lira  sin  x  =  l,  lim  cos  a:  =  0, 

we  have  1),  by  277,  6. 

2.  Similarly, 

i2  lim  tan  x  =  —  oo. 

298.  1.  lime^=l. 

x=0 

This  follows  at  once  from  172. 


2. 

lim  e^  =  e". 

For,  let 
Then,  by  289, 

x=  a  +  u. 
lim  e^  =  lim  e'^+«  =  e'^  lim  «« 

x=a                  «=0                                «=0 

=  g«,  by  1. 

186  LIMITS   OF   FUNCTIONS 

3.  lim  e^  =  +  QO . 

This  follows  at  once  from  169. 

4.  lim  e-^  =  0. 
For,  let  _  H 

u 

Then  lim  e^  =  i2  lim  -,  by  290,  2  ; 

x=—a3  M  =  0       - 

=  0,  by  277,  5. 

5.  Obviously,  as  in  1,  2, 

lim  a^  =  a^". 

I  =10 

1 

e^-1 


6.  /(^)  =  -T 

e^  +  1 

i?  lim/(a;)  =  +  1,         L  lim/(2;)  =  -  1. 

x=0  1=0 

299.  1.  Let 

lim/(a;)  =  ?;.  ■J?  >  0. 

lim(/(2:)y  =  7;^ 

z=a 

This  follows  directly  from  171. 
2.  In  F*(a),  Zef  /(a;)  >  0.     Let 
\\mf(x)  =  0. 

x  =  a 

Then 

lim(/(a;)y=0,  /i>0. 

=  1,  ^=0. 

=  +  00.        /t<0. 

300.  1.  lim  log  a;  =  log  a.       a>0. 

x  =  a 

This  follows  at  once  from  178. 


EXAMPLES   OF   LIMITS   OF   FUNCTIONS 

2.  lim  log  a;  =  +  00 . 

This  follows  at  once  from  179. 


3. 

For,  set 


Then 


Ji  lim  log  x=  —  oo, 

x  =  0 

1 

X=  -' 

u 


R  lim  log  X  =  lim  log  -  =  —  linj  log  m  =  —  oo . 


4.  Let 
Then 


lim/(a;)  =  t;  >  0. 
lim  log/(a;)  =  log  r)  =  log  lim/(a;). 

x=a  x=o 

This  follows  at  once  from  178. 


187 


301.    1. 


T      Sin  a:      ^ 
lim =  1. 


z=0         X 

From  geometry,  we  have 

Area  0^C<  Area  05a<  Area  OBD. 
Hence,  for  0  <  x  <  7r/2, 

^  sin X  cos  x<^  x<^  tan  x ; 


or 


As 


Set  in  1), 


Then 


cos2;< 

Sill  X 

1 

cos  a; 

i^lim 

x=0 

cosa;  = 

R  lim     ^     = 

x=0  COS  X 

1, 

277,3 

1 

Blim 

x  =  0 

X 

1. 

sin  X 

x  = 

—  u. 

i21im 

x  =  0 

X 

i  lim 

u  =  0 

u 

sin  a; 

sin  w 

=  1. 


(1 


(2 


188  LIMITS  OF  FUNCTIONS 

From  1),  2)  we  have 

lim  — =  1. 

x=o  sin  X 

Whence,  by  277,  2, 

T     sin  a;      -, 
lim =  1. 

x  =  0         X 

2.   From  1  we  have  readily 


For, 


1 .     sin  ax      a  in 

lim—- =  -•         5  9^0. 

x=o      bx         0 


sm  ax      «    sin  ax 


hx         h       ax 
a    sin  u 


setting 


h       u 


u=  ax. 


302.  lini*?:IL^=l.  (1 

x=0  X 


For, 
But 


tan  X  _  sin  x       \  (2 

X  x       cos  X 

1 .     sin  X      ^    -,.        1         -, 
iim =  1,  lim =  1. 

x=o      X  1=0  cos  X 


Thus,  passing  to  the  limit  in  2),  we  get  1),  by  277,  2. 

«fto                         T  •     sin  (x  +  K)  —  sin  x  ,., 

303.  lim ^^ — -—^ =  cos  X.  (1 

ft=o  h 


For, 


sin  (x-\-K)  —  sin  x  _'2  cos  (ar  +  ^  A)  sin  \  h 
l  h 


^^^^  lim  cos  (a;  +  1  A)  =  cos  a: ; 

ft=0 

lim  2li5_y:=  lim  ^1^  =  1. 
Passing  to  the  limit  in  2),  we  get  1). 


(2 


304.    1. 

For, 

2. 
For, 

305     1. 

Here 
while 


EXAMPLES   OF   LIMITS  OF   FUNCTIONS 


1-     1  —  cos  a;_  1^ 

05=0        x^  2 


1  —  cos  a:  _  2  siii^  \  x  _  1/sin  \  x^ 
x^        ~        x^        ~^    \x    )' 


189 
(1 


lim 

a:=0 


tan  X  —  sin  x  _  1 

^3  -^ 


tan  x  —  sin  x      tan  x  \  —  cos 


3? 


lim  e""'  cos  re  =  0. 


(1 


lim  e-^  =  0,  by  298,  4, 
lim  cos  x 


does  not  exist.     We  cannot,  therefore,  apply  the  theorem  277,  2, 
that  the  limit  of  the  product  is  the  product  of  the  limits. 
We  therefore  proceed  thus : 

—  e~'  <  e'"  cos  X  <  e~^. 

Apply  now  277,  3.     This  gives  1). 

2.  We  may  see  the  truth  of  1)  geometrically. 

Let 

«/l=e"^  ?/2  =  cosa:. 


and 


y  =  e  "  cos  rr  =  y^y^. 


Let  us  draw  the  graphs  (7,  C  of  i^/j. 

To  get  ^,  we  multiply  y^  by  the 
factor  y^,  which  takes  on  all  values 
between  —  1  and  + 1.  Thus  y  oscil- 
lates between  the  curves  (7,  C. 

As  (7,  C  approach  nearer  and  nearer  the  a;-axis,  the  amplitude 
of  the  oscillations  converges  to  0. 


190  LIMITS   OF   FUNCTIONS 

Tlie  Limit  e  and  Related  Limits 
306.    1.   We  saw  in  110  that 

lim 


imfl  +  -)=e;  7i=l,  2,  3,  •.•  (1 


Let  us  consider  now  the  more  general  limit 

1^ 


lim    1  + 

Each  X  will  lie  between  two  integers  w,  w  +  1 ;  or 
n  <x  <n-\-\. 


Then 


1+-^H-->1  +  ^, 
n  X  n-\-\ 


and 


But 


and 


Thus  3),  4)  give  in  2), 

^n  +  1 
Now,  by  1),  ,        ^v  ,  .     .     , 

lim(l  +  -)   =l™(l  +  -l3)     . 

liml  +  -   =lim = — =1. 

w+  1 
Hence  5)  gives 


THE  LIMIT   E   AND   RELATED   LIMITS  191 

307. 

lira 


liin('l+iy  =  e.  (1 


For,  let 
Then 


x  =  —  u. 


(^-r=(-r=(^-.4iT 


=(^-D(^-^)>  <^ 


where 


But,  when  a;  =  —  oo,  m  =  +  oo,  and  hence  t)  =  +  go. 
^^  Umfl  +  ^')  =  1,    Hmfl+iY=e, 

we  get  1),  on  passing  to  the  limit  in  2). 
308.    From 


lira(l+-     =e, 

x=+oo  V           xJ 

we  get,  setting 

1 

x  =  -, 
u 

1 

R\\m(l  +  uy=e. 

«=0 

From 

lim(l  +  -)   =e, 

we  get,  setting 

1 

x  =  --, 
u 

1 

i  lim  (1  +  uy  =  e. 

«=0 

From  1),  2)  we 

have 

1 

lim  (1  +  xy  =  e. 

x=0 

(1 


(2 

(3 


192  LIMITS   OF   FUNCTIONS 

309. 


For, 

where 
But 


1 
lim  (1  +  xuy  =  e^.  (1 

1  J_ 


V  =  ux. 
1 
lim  (1  +  vf  =  e. 


ti=0 


310.  l™ 

1=0 

For, 


Hence,  by  299,  ,.  _ 

•^  liiu  y  =  e^. 

log:  (1  +  ^')  _  -I 

X 

lim  log(^  +  ^0  _  |i„^  Yog  (1  +  rc)^ 

x=0  X  1=0 

1 

=  log  lim  (1  4-  xy,  by  300,  4) 

1=0 

=  log  e,  by  308,  3) 
=  1. 

a*  —  1 
311.  lim =  log  a.  a>0.  (1 

x=o       a;  ° 

Set 

M  =  a-^  —  1. 
Then 

lim  M  =  0, 

a:=0 

by  298,  5.     Then,  by  292, 

j.^log(l  +  »)^j.^log«' 

,4=0  M  a^  a^—  1 

=  1,      by  310. 
But 

log  a''  =  x  log  a. 
This  in  2)  gives  1). 


DEFINITIONS   AND   ELEMENTARY  THEOREMS  193 

312.  ,.^(l  +  .)>-l^ 


From  310  we  have 

lini^Ml+^=l. 
Let 

u  =  (i  +  xy  —  i.       n^o. 

Then,  by  299, 

lim  w  =  0. 

x=0 

Hence,  by  292,  we  get  from  2) 

^.^log(l+^^ 

=^  a  +  a^y-i 


(2 


or 

since 

But,  by  310, 


=^  (1  +  ^y 

(l  +  a:y-l  „ 

log  (1  +  ^) 

log(l  +xy=fM  log(l  +  x). 


lim  - — ,-,  ^    .  =  1.  C4 

log(l  +  a;) 

Now 

(1  +  a;)*^  -  1  _  (1  +  xy  -  1  X 

log(l+a;)  ~  X  log(l  +  a;) 

Passing  to  the  limit  and  using  4),  we  get  1)  for  the  case  that 
fjL^O.     The  case  that  ft  =  0  is  self-evident. 


FUNCTIONS  OF  SEVERAL  VARIABLES 

Definitions  and  Elementary  Theorems 

313.  For  the  sake  of  clearness,  we  have  treated  first  the  limits 
of  functions  of  a  single  variable.  We  consider  now  the  limits  of 
functions  in  m  variables.  The  extension  of  the  definitions  and 
results  of  the  preceding  sections  is,  for  the  most  part,  so  obvious 
that  we  shall  not  need  to  enter  into  much  detail.  Should  the 
reader  have  trouble  with  the  case  of  general  m^  let  him  first  sup- 
pose m  =  2  or  3,  when  he  can  use  his  geometric  intuition  as  a 
guide. 


194  LIMITS  OF  FUNCTIONS 

314.  In  the  case  of  a  single  variable,  we  have  seen  how  useful 
the  ideal  points  ±  oo  proved.  In  the  treatment  of  limits  of  func- 
tions of  several  variables,  we  shall  find  it  extremely  advantageous 
to  adjoin  an  infinity  of  ideal  points  to  dim  as  follows : 

Let  A  =  aj,  a^^  a^,  •••  be  an  infinite  sequence  of  points  in  QfJ^. 
Let  .  ,     ,, 

lim  a's  =  «!,  •••  lim  a/"''  =  a^; 

^v  •••  ^mi  finite  or  infinite. 
We  say  the  limit  of  the  sequence  A  is 

and  write 

a  =  lim  a„. 

If  any  of  the  coordinates  of  «  are  infinite,  we  say  a  is  an 
infinite  point.     This  fact  may  be  briefly  denoted  by 


the  symbol  oo  being  without  sign. 

There  is  no  point  in  dim  corresponding  to  an  infinite  «.     We 

therefore  introduce  an  infinite  system  of  ideal  points,  one  for  each 

complex,  ^_ 

a^,a^,'"a^,  (2 

in  which  one  at  least  of  the  symbols,  a^,  is  ±oo.     Such  ideal  points 
we  represent  also  by  1),  and  q 

call  the  m  symbols,  2),  their     _oo< t^ ^oo 

coordinates.       If    we    employ 

the     graphical    representation      ~°°^  ^  ^'^'^ 

of  231,  we  suppose,  according  to  275,  that  each  axis  is  terminated 
by  the  ideal  points   +qo   and  —  oo. 

Thus,  any  complex  of  m  points,  one  on  each  axis,  such  that  at 
least  one  of  these  m  points  is  an  ideal  point,  is  the  representation 
of  an  ideal  point  in  9?^. 

The  system  of  points,  formed  of  9?^  and  the  ideal  points,  we 
denote  by  9?^. 

These  ideal  points  are  also  called  points  at  infinite/. 


DEFINITIONS   AND   ELEMENTARY  THEOREMS  195 

315.  1.  The  domain  of  an  ideal  point  a  =  (a^  '"<^m)  is  the  aggre- 
gate of  points  X  =(,x^^"X„^'),  whose  coordinates  lie  respectively  in 
the  domains 

It  may  be  represented  by  Dp^...p^(a). 

Example.     Let  m=4,  and  a=(  — oo, 

1,2,  4-00).     The  domains  in  which  the     _oo.«— —i i2_ >.'ioo 

coordinates   xi,   Xo,    xg,   Xi    range,   are  ''i                ^ 

marked  heavy  in  the  ligure.  -oo<:  i — -^    •     ^ >  ^-oo 

Here  px  is  an  arbitrarily  large  nega-      -cx3< f^     \      '     ii         ,  i  m 

tive  number;  p2  and  pz  are  arbitrarily      _^^ o      »  a 

small  positive  numbers;  pi  is  an  arbi-  Pt 
trarily  large  positive  number. 


2.  The  points  of  an  aggregate  A^  which  lie  in  Dp^...p^{a'),  a  being 
a  point  at  infinity,  form  the  vicinity  of  a,  for  that  aggregate.  We 
represent  it  by 

316.  1.  Let  y  =f(x-^'--Xj^)  be  defined  over  a  domain  D.  Let 
A  =  aj,  a2,  •••  be  a  sequence  of  points  in  i),  and  let 

lim  a„  =  a.  a  finite  or  infinite. 

If 

lim/(a„)  =  77,         7)  finite  or  infinite. 

is  always  the  same  however  A  be  chosen,  a  remaining  fixed  and 
a„  rjfc  a,  we  say  r]  is  the  limit  of  y  for  x=  a;  and  write 

r]  =  \imf(x^...x^'), 
or,  more  briefly, 

7]  =  lim/(a:j  ...a;^),  or  77  =  lim/(a;); 
or, 

2.  Just  as  in  the  case  of  a  single  variable,  we  can  show  that  this 
definition  is  equivalent  to  the  following : 

7?  =  li  m  /  (a^j  •  •  •  x^') ,         7]  finite  or  inf. 

x=a 

when,  taking  D{r)^  arbitrarily  small,  there  exists  a  vicinity  V*(^a), 
such  that  y  remains  in  D(j]^  when  x  is  in  V*(^a).     See  278-283. 


196  LIMITS   OF   FUNCTIONS 

317.  1.   The  theorems  of  277   and   284  hold  for  functions  of 
several  variables  as  well  as  for  a  single  variable. 

2.   The  generalized  theorem  of  292  may  be  stated  thus ; 

Let  w^  =  (/)j(a:j...a:„),  •  •  •  it„,  =  <^,„ (a^i •  •  •  a:„) ; 

and 

lim  Mj  =  5j,  •••  lira  m„j  =  h^. 

x=a  x=a 

Let 

and 

lim  y  =  Tf). 

Letu^b  in  F*(a).      Then 

lim  y  =  7], 

x—a 

Here  a,  J,  77  may  he  finite  or  infinite. 

The  demonstration  is  perfectly  analogous  to  that  in  292. 

A  Method  for  Determining  the  Non-Existence  of  a  Limit 

318.  To  determine  whether 

7]  =  \\vQ.f(x-^'-'X^,         a,  77  finite  or  inf. 

even  exists,  is  often  a  difficult  matter.  The  following  simple  con- 
sideration analogous  to  273,  2,  3  will  sometimes  show  very  easily 
that  77  does  not  exist.  Let  Whe  some  partial  vicinity  of  a  exclud- 
ing a.  We  may  denote  the  limit,  when  it  exists,  of  f(x-^^--'X^  for 
x  =  a  when  x  is  restricted  to  W  by 

^=\imf(x^-"X^). 

w 

Then  K,  must  exist,  finite  or  infinite;  and  however  Tf  is  taken, 
we  must  have 

Thus,  in  case  ^  does  not  exist,  or  is  different  for  different  TF's, 
we  know  that  t]  does  not  exist. 


DETERMINING   THE     NON-EXISTENCE   OF  A   LIMIT      197 

We  ask,  does 

lim/ 

i,y=0 

exist  ?    As  partial  vicinity  of  the  origin,  take  points  on  a  line 

L;  y  =  ax.  X  ^  0. 

Then 


lim/(a;,  y)  =  lim       /^^        =  -^^ 

i  x=o  x-(l  +  a^)     1  +  a^ 


which  varies  with  a  ;  J.e.  with  L. 

Hence  the  limit  in  question  does  not  exist. 

320.    Ex.2.  2 

Ax,y)  =  -^. 
x^  +  2/* 
Does 

lim/  a 

x,y=0 

exist  ? 

If  we  take  as  partial  vicinity  of  the  origin  points  on  the  line 

L  ;  y  =  ax, 

we  get  2 

lim  f(x,  y)  =  \imx-       "  ^  .^  =  Q.  (2 

L  x=o       1  +  a*x^ 

Thus,  however  L  is  chosen,  the  limit  2)  is  always  the  same.  We  cam  ot,  how- 
ever, infer  that  the  limit  1 )  exists,  since  our  method  only  shows  the  non-existence 
of  the  limit. 

Instead  of  the  family  of  right  lines  L,  let  us  take  a  family  of  parabolas 

P ;  y-  =  ax. 

lim/(a;,  ?/)=lim        ^^  ^ 


x=ox^(l  +  a^)      1  +  a^' 


which  varies  with  the  particular  parabola  chosen. 
Hence  the  limit  1)  does  not  exist. 


321.    Ex.3. 

Does 


f(x,y)  =  log- — -.      x,y<a. 


a  -y 


lim/ 

x,y=a 


exist  ? 

Let  X,  y  lie  on  the  line 
L;  a  —  x  =  \(a  —  y).  X>0. 


Then 
Hence 
which  varies  with  X. 


/(x,  y)=  logX. 
lim/(x,  y)=  logX, 


198  LIMITS   OF   FUNCTIONS 

Iterated  Limits 
322.    1.  Let /(zj  •  •  •  a;^)  be  defined  over  some  domain  D\  and 
let  a  =  (a^"-aj^. 

Then  lira  f(x^  "•x^^=  /^ 

will  be  in  general  a  function  of  all  the  variables  except  x^^.      Also 
lim  /^  =  /^.,  =  lim  •  lim  f(x^  •  •  •  x^) 


'2       '2         '1       '1 


will  be  in  general  a  function  of  all  the  variables  except  x^^,  x^^. 
Continuing,  we  arrive  at 

lim  •  •  •  lim  •  lim  /(a^j  •  •  •  a;^),  s  <  w,  (1 

«t»=<»t,         ="l2=''l2     ^'l="h 

which  is  in  general  a  function  of  all  the  m  variables  except 
Xi^t  a^ij^  ' '  *  x^^. 

Limits  of  the  type  1)  are  called  iterated  limits. 

In  1),  we  pass  to  the  limit  first  with  respect  to  a^^^,  then  with 
respect  to  x,^.,  then  with  respect  to  x,^.,  etc. 

A  change  in  the  order  of  passing  may  produce  a  change  in  the 
final  result. 

2.  Iterated  limits  occur  constantly  in  the  calculus ;  for  example, 
in  partial  differentiation,  differentiation  under  the  integral  sign, 
double  integrals,  improper  integrals,  and  double  series.  The 
treatment  of  these  subjects  by  the  older  writers  on  the  calculus 
is  faulty,  as  we  shall  see,  because  they  change  the  order  of  passing 
to  the  limit,  without  a  careful  consideration  of  the  correctness 
of  such  a  step. 


323.    Ex.  1.  limlim^-:^  =  limf^:ii^^  =  -l. 

i,=o  x=o  a;  +  2/      y=o\  y  J 

lim  lim  5^^  =  lim  f -"^  =  + 1. 

x=0   y=0  X  +  y       1=0  \x/ 

The  two  limits  are  thus  different. 

Ex.  2.  lim  lim  hz^  =  _  i. 

y=s)  1=00  1  +  xy 


lim  lim  5^^^^  =  +  1. 

I=«,   y=0    1  +  SCy 


The  two  limits  are  thus  different. 


UNIFORM   CONVERGENCE  199 

324.    The  following  is  a  case  where  a  change  in  the  order  of 
passing  to  the  limit  does  not  change  the  result. 

Let  T       /. 

lim  f{x,  y^=r],  7}  finite  or  inf.  (1 

x=a,  y=b 

\\VLif(x,y)  =  g(x),       forO<\x-a\<a,  (2 

lim f(x,  y')=h(iy'),       for  0<\y -b\<a.  (3 


Then 


lim  ^(a;)  =  lim  k(^y}  =  rj.  (4 


V=6 

Let  T)  he  finite.     From  1)  we  have 

r]  —  e<  f(x,  2/)  <  ?7  +  e,        in  V&*(^a,  5). 

In  this  relation,  pass  to  the  limit  x  =  a;  then 

■q  — e<g{x)<7] +  €,    /or  0<|a;  — a|<S. 
Hence 

lim  gQx)  =  rj.  (5 

Similarly, 

lim  A(y)  =  7).  (6 

From  5),  6)  we  have  4). 
Let  97  =  +  00.     Then  from  1) 

f(x,y')>a,         in  F,*(a,  6). 

Passing  to  the  limit,  for  2:  =  a,  we  have 

g(x)^a,        forQ<\x-a\<h. 
Hence 

lim  g(pc)  =  -}-  CO  =rj. 
Similarly, 

lim  h(x^  =  -\-  Qc  =7]. 

These  two  equations  give  4). 

Uniform  Convergence 

325.  1.  A  notion  of  utmost  importance  in  modern  mathematics 
is  that  of  uniform  convergence.  Let /(a^j  •••  «^;  ^j  •••  f„)  be  a 
function  of  two  sets  of  variables,  x-^  •"  Xj^  and  t^  •••  t^. 


200  LIMITS   OF   FUNCTIONS 

Let  /  be  defined  when  x  =  (rcj  •  •  •  a;^)  ru  ns  over  a  domain  i), 
and  i=  (^1  •••  O  runs  over  A. 
For  each  x  in  D,  let 

limf(x^  •••  x,„;  f^  •••  Q  =  g(x^  •••  x„,). 

i  =  T 

Then  for  each  e  >  0  and  each  x  in  D  there  exists  a  S'  >  0,  such 

|/-^|<e  (1 

for  any  t  in  F^'*(t). 

Evidently  if    1)  holds   for   8',  it   holds  for   any  S",  such   that 

0<B"  <B'.     Of  all  the  values  8'  for  which  1)  holds  at  x,  let  B  be 

the  maximum.     Then  for  a  given  e,  S  is  a  well-defined  function 

of  X.     In  D,  let 

Mm  0  =  Oq. 

Then  Bq  ^  0.  If,  however  small  e  is  taken,  the  corresponding  Bq 
is  >  0,  we  say  /  converges  uniformly  to  g  in  D ;  or  is  uniformly 
convergent. 

Hence,  if  /  is  uniformly  convergent  in  Z>,  there  exists  for  each 
e  >  0  a  2  >  0,  such  that 

for  any  t  in  F^*(t).     Moreover,  one  and  the  same  norm  B  suffices 
for  all  the  points  of  D,  e  being  the  same. 

The  central  idea  of  this  case  of  uniform  convergence  may  be 
clearl}^  if  somewhat  roughly,  brought  out  by  saying  that  if  the 
convergence  is  uniform  the  norms  B  for  which  1)  hold,  e  being 
small  at  pleasure,  but  then  fixed,  do  not  sink  below  some  definite 
positive  number,  when  x  ranges  over  D. 

2.  These  considerations  may  be  extended  to  the  case  that  t  is 
infinite  ;  we  therefore  define  as  follows : 

The  function  /(a^j  -"  x^;  ^^  •••  ^„)  converges  uniformly  to 
g(^x■^^  ••'  a;,„)  in  D  as  t  =  t,  t  infinite;  when  for  each  e>0,  there 
exists  a  set  of  norms  Pi  '•-  Pn-,  such  that  for  any  x  in  i), 

|/(a^l    •••  ^ml    h   •'•  tn)  -ffC^l   ••'  ^m)|<e 

in  Fp^...p„(T). 

In  this  case  of  uniform  convergence  we  may  say:  the  norms  p 
corresponding  to  infinite  coordinates  of  t  cannot  become  infinitely 


UNIFORM   CONVERGENCE  201 

great,  and  the  norms  corresponding  to  finite  coordinates  of  t  can- 
not become  indefinitely  small  as  x  ranges  over  i>,  for  any  given 
value  of  e. 

3.  When  f{x^  •••  x„^\    t^  •••  ^„)  converges  uniformly  in  D  to 
g(^Xj^  •••  a;^),  we  denote  this  fact  by 

YimfQx^  ■■■  x„^;  t^  ■■■  t^^  =  g(x^  ...  a;^),    uniformly. 

4.  If  /=  0  uniformly  in  i),  we  may  say  it  is  uniformly  evanescent 
in  D. 

326.  Ex.  1.  , 

Z>=(0*1).        A=(-h,h).        h>0. 

Evidently  for  any  x  in  D 

lim/(x,  t)  =  ^=g(x). 

(=0  * 

But  f(x,  t)  does  not  converge  uniformly  to  rj{x)  in  D.     For  if  it  did,  for  each 
e  >0  there  must  exist  a  S  >  0,  such  that 

R  =  \f(x,t)-(J{X)\=        1^1      ,<e  (1 

x\x  +  t\ 
for  any  t  in  Fj*(0)  and  any  x  in  I>. 

Now  obviously,  t  being  fixed,  ^  can  be  made  as  large  as  we  choose  by  taking  x 
near  enough  0.     Hence  B  does  not  satisfy  1)  as  x  ranges  over  D. 

In  fact,  as  is  seen  at  once,  in  order  to  have  i?  <  e,  it  is  necessary  to  take  5 
smaller  and  smaller  as  x  approaches  0.     In  this  case  then 

Min  5  =  0,         in  D. 

327.  Ex.  2. 

x  -\-  t 

D=(n,b),   0<a<6.         A=(-h,h),    h>0. 

This  example  is  the  same  as  Ex.  1,  except  D  is  different. 

As  before  ^ 

lim/(x,  t)  =  -=g(x). 
t=o  X 

But  now/(x,  t)  converges  uniformly  to  g{x)  in  D. 
In  fact,  in  V^*(P')  - 

i?< ,         5<a, 

«(a  —  5) 

wherever  x  is  in  D.     But  we  can  take  5,  such  that 

a{a  —  5) 
Then 

B<€ 

tor  any  t  in  V^(())  and  any  x  in  2>  ;  i.e.  f  converges  uniformly  in  D. 


202  LIMITS  OF   FUNCTIONS 

328.   Ex.  3.  , 


Here 


Hence  if  we  set 


we  have 


(1  +  X2)' 

D  =  {—a,a).        A=(a,  +Qo). 
lim/(x,  t)=l  +  x^,        x^O. 

t=CO 

=  0.         X  =  0. 
g(x)—0,        for  x  =  0 

=  1  +  x2,        for  X  ^fc  0, 
lim/(x,  t)=g{x). 


However,  /  does  not  converge  uniformly  to  g  in  D.     For,  when  x  ^  0, 

1 


B  =  \f(x,t)-g(x)\  = 


(1  +  x2)« 


This  shows  that  as  x  approaches  0,  it  is  necessary  to  take  t  larger  and  larger  in 
order  that  i?  <  e. 

There  is  thus  no  norm  p,  such  that 

for  each  t>  p,  and  any  x  in  Z>. 
In  this  case,  then, 

Maxp  =00. 


Remarks  on  Dirichlefs  Definition  of  a  Function 

329.  The  definition  of  a  function  given  in  189  and  230  does  not 
depend  at  all  upon  an  analytic  expression  for  the  function. 

At  first,  the  reader  who  has  been  used  only  to  functions  defined 
by  analytic  expressions,  may  be  inclined  to  regard  functions  not 
thus  defined  as  only  pseudo-functions,  or  at  least  of  little  impor- 
tance. 

This  attitude  of  mind  must  be  overcome.  In  the  first  place,  in 
certain  parts  of  mathematical  physics,  e.g.  the  potential  theory,  it 
is  of  great  importance  to  be  able  to  assign  values  to  a  function  at 
pleasure,  totally  disregarding  the  question  of  an  analytic  expression 
for  it. 

Secondly,  as  the  reader  advances,  he  will  find  that  many  func- 
tions which  he  might  well  believe  have  no  analytic  expression,  do 
indeed  have  very  simple  ones. 

We  give  now  a  few  examples  of  such  functions. 


DimCHLETS   DEFINITION   OF   A   FUNCTION 


203 


330.  1.  For  x>0  let  7/  =  l. 

For  x  =  0  let  t/  =  Q. 
For  a;  <  0  let  ?/  =  —  !. 

The  graph  of  this  function  is  given  -to 
in  the  figure.  ^^_^ 

An  analytic  expression  of  ^  is 

2 
y  =  —  lim  arctg  (jix). 

TT    71=00 

This  function  is  much  used  in  the  Theory  of  Numbers.     We 
shall  call  it  signum  x  and  denote  it  by 

y  =  sgn  X. 

When  w,  V  Tib  0  an  equation 

sgn  u  =  sgn  V 

simply  means  that  the  sign  of  u  is  the  same  as  that  of  v ;  while 

sgn  w  =  +  1 

is  only  another  way  of  saying  that  u  is  positive  etc. 

331.  For  x=^0  let  y  =  1. 
For  a:  =  0  let  ?/  =  0. 


Its    graph    is   indicated   in    the  -con- 
figure. 

An  analytic  expression  of  i/  is 

nx 


y  =  lim 


\-\-nx 


332.    Fora;=0,    ±1,   ±2,  •••  let  y=0. 

For  ?i<2:<r2,  +  l,  ?/  =  (—  1)''.       n positive  integer  or  0. 
For  —(n  +  l)<x<  —  n,  y  =  — (— l)^ 

The  graph  of  y  is   indi- 
cated in  the  figure.  ^^       

An  analytic  expression  of 

y  is  "  '       ^ 

T     (1  ^- sin  7ra;y  -  1  

y  =  lira  ^^-— ! — : ^ -•  — 

n=«  rl  4- sin  Tra;')" -f  1 


In  fact : 


Example. 
Then 


204  LIMITS   OF   FUNCTIONS 

333.  Let  /(a;),  g(x)  be  two  different  functions,  defined  over 
51  =  (0,  +  oo).  The  inexperienced  reader  might  well  believe  that 
we  cannot  form  an  analytic  expression  which  represents  f(x)  in 
one  part  of  91,  and  g(x)  for  another  part.  Such  an  expression  is, 
however,  the  following : 

V  =  lim     -^    '  '  i)\   /. 
^       „==c         a;"  +  1 

for  x>\,  y=f{x), 

for  0<a:;<l,  y  =  g(x), 

for  rr=l,  ^  =  H/(1)  +  K1)|. 

/(x)  =  x^,  g^Cx)  =  COS  2  TTX. 
2/  =  x^,  for  x>  1, 
=  cos  2  TTX,  for  0  <  X  <  1. 

334.  1.  For  rational  x,  let  y—a\  for  irrational  a;,  let  y  =  h, 
where  a,  5  are  constants.  This  function  was  introduced  by 
JDirichlet. 

In  any  little  interval,  y  jumps  infinitely  often  from  a  to  b  and 
back.  It  seems  highly  improbable  that  such  a  function  should 
admit  a  simple  analj^tic  expression  ;  yet  it  does. 

We  have  already  seen  that  sgn  x  admits  a  simple  analytic 
expression. 

Consider  now 

y  =  (J  +  (5  —  a)  lim  sgn  (sin% !  irx) .  (1 

For  any  ratiojial  x.,n\x  finally  becomes  and  remains  an  integer. 

Hence  sinwiTra;  =  0  for  sufficiently  large  n. 

Hence  y=  a  for  any  rational  x. 

For   an    irrational  a;,  n\x  never   becomes  an  integer.     Hence 

^\v?n\'Kx  lies  between  0  and  1,  excluding  end  values. 

Therefore 

sgn  (sin^  n !  irx^  =  1 ; 

and  for  any  irrational  x^  y=h. 


UPPER   AND   LOWER   LIMITS 


205 


Thus  1)  is  an  analytic  expression  of  Dirichlet's  function. 
The  reader  should  note  that  it  is  utterly  impossible  to  intuition- 
ally  realize  the  graph  of  this  function. 

2.   Similarly,  we  see  that 

y  =/(^)  +  (^(3^)  — /(^)  )  lim  sgn  (sin^  n !  irx) 

71=00 

equals  /(^)  when  x  is  rational, 

and  equals  g{x)  when  x  is  irrational. 

335.    A  remarkable  function  is  the  following.     We  shall  call  it 
Cauchy^ s function,  and  denote  it  by  0(x'),  viz.: 

_j_ 
C(jc)  =  e  ^\  for  x^O, 

=  0,  for  x=  0. 
As  a  limit,  we  can  write  it 


or 


(7(2;)  =  lime  ■^'+"' 

u=0 


C(x)=  lim  e 


Its  graph  is  given  in 
the  figure.  Its  peculiarity 
is  its  remarkable  flatness 
near  the  origin. 


Upper  and  Loioer  Limits 

336.    1.   Let/(a;j  •••  x„^=f(x)  be  defined  over  D. 

Let  a  be  a  limiting  point  oi  D;  a  and  D  may  be  finite  or  infinite. 

Let  A  =  «j,  a^,  ••  •  be  a  sequence  of  points  in  D  whose  limit  is  a, 

such,  however,  that 

L  =  lim/(a„) 

n=ao  J 

exists,  finite  or  infinite.     There  are  an  infinity  of  such  sequences. 


206  LIMITS   OF   FUNCTIONS 

For  all  such  sequences,  let 

X  =  Min  L,         11=  Max  L. 

These  are  called  respectively  the  lower  and  upper  limits  of 
f(x^  •  ■  •  x^')  at  a;  we  write 

X  =  lim  inf /(a^i  •  •  •  a;^)  =  lim/=  lim/, 

x=a  x=a 

fi  =  lim  supf(x^  '■•  x^  =  lim/=  lim/. 

The  lower  and  upper  limits  X,  /t  may  be  infinite. 

2.  When  dealing  with  functions  of  a  single  variable,  we  can 
have  right  and  left  hand  upper  and  lower  limits^  by  considering 
only  values  of  a:;  >  a,  or  <  a,  respectively. 

Then 

R  lim  sup/(a;)  =  lim  sup/(a:)  =  \\m.f(x) 

x=a  1=0+0  1=0+0 


=  i21im/=/(a  +  0) 

all  denote  the  right  hand  upper  limit  of  f(x)  at  a.     A  similar 
notation  is  employed  for  the  left  hand  limits. 


337.  EXAMPLES* 

1. 


.    1 

y  —  sin  -  • 

X 


limy  =  — 1,  lim?/  =  +  l. 

x=0  a;=0 

2.  ?/=  (l-x2)sini. 

^         X 

lim2/  =  — 1,  Iim?/=+l. 

x=0  1=0 

3.  ?/=(!  +  a;2)  sin  -  • 
lim  y  =  —  1.  Tim  y  =  +  1. 

a=0  z=0 


«"(  a  +  sin  ^— ^  +  6  +  sin  — i— 
4-  .  .j,  =  lim-^ ^^lli ^Ili. 

n=o  1  +«" 

See  333. 

*  The  reader  will  do  well  to  roughly  sketch  the  graph  of  these  functions. 


We  find : 


UPPER   AND   LOWER   LIMITS  207 


y  z=  a  +  sin ,  for  x  >  1. 

X  —  1 


Hence 


y  =  b  +  sm—^,  /or  0<x<l. 

X  —  1 

B  lim  y  =  a  +  1;  L  lim  y  =  b  +  1; 

X=l  1=1 

B  IJrn  y  =  a  —  l;  L  lim  y  =  6  —  1. 

x=l  x=l 

338.   1.    Let  \,  /Lt  6e  the  lower  and  upper  limits  of  f(x-^  '"^ni)  ^^ 
x  =  a.     Then  there  exists  for  each  e  >  0  a  S  >  0,  such  that 

\-€<f(x^---x^}<fi  +  €,         mVs*(a'), 

a,  finite  or  infinite. 

For,  in  the  contrary  case  there  exist  sequences  A  =  ap  tig'  "' 

such  that  ,.       -      ^ 

hm/(a„)<\, 

or  lim/(a„)>/x. 

2.  Obviously  we  have  the  following  : 

Let  A,,  fi  he  the  loiver  and  upper  limits  of  f{x-^  •  •  •  x^  at  a.     There 

exist  tivo  sequences  A  =  a^,  a^,  •••,  B  =  h^.,  b^,  •••  whose  limits  are  a, 

such  that 

lim  /(a„)  =  \,    hm  /(6„)  =  ^. 

3.  Since  the  maximum  and  minimum  of  a  variable  exist,  finite 
or  infinite,  we  have  : 

The  upper  and  lower  limits  of  a  function  always  exist  finite  or 
infinite.     If 


Hrn/=lim/  =  i, 


then 


lim  f  =  l. 


CHAPTER   VII 
CONTINUITY  AND  DISCONTINUITY  OF  FUNCTIONS 

Definitions  and  Elementary  TJieoreyns 

339.  1.  Let  f(x-^'--  x„^^  be  defined  over  a  domain  D.  Let 
a  =  {a^  •'•  a^  be  a  proper  limiting  point  of  I).     If 

lim  /(a^i  •  •  •  a;,„)  =  /(«!  •  • .  a^) ,  (1 

the  function/ is  continuous  at  a.      In  words:   if  the  limit  of  f  at  a 
is  the  same  as  the  value  of  f  at  a,  it  is  continuous  at  a. 

The  reader  should  observe  that  a  is  not  only  a  limiting  point  of 
D,  but  that  it  lies  in  D. 

2.  The  condition  1)  may  be  expressed  in  the  e,  8  notation,  giv- 
ing the  following  definition  of  continuity :  f{x-j^  •••  a;„j)  is  continuous 
at  a,  if  for  each  e  >  0  there  exists  a  S  >  0,  such  that 

Ifi^i  •••^m)- /(«!  •••«,„) I  < e,         in  Fa(a) . 

3.  A  function  which  is  continuous  at  all  the  proper  limiting 
points  of  D  is  said  to  be  conthiuous  in  D.  We  suppose  that  D 
has  at  least  one  proper  limiting  point. 

4.  Consider  the  function 

n^        X  _      xy  at  points  different  from 

J\^i  I/)—  ~T~: — 9'         ^7         •   • 
x^  +  y^  the  origin. 

=  0,        at  the  origin. 
We  saw,  319,  that 

lm\f(x,  y') 

1=0,  y=n 

does  not  exist.     Thus  /  is  not  continuous  at  the  origin. 

208 


DEFINITIONS  AND   ELEMENTARY  THEOREMS  209 

At  the  same  time  /  considered  as  a  function  of  x  alone,  or  con- 
sidered as  a  function  of  y  alone,  is  continuous. 

This  example  illustrates,  therefore,  the  fact  *that  because 
/(.T J  •  ■  •  a:,„)  is  a  continuous  function  of  each  variable  separately, 
we  cannot,  therefore,  assert  that  /  considered  as  a  function  of 
rTj  •  •  •  x„^  is  continuous. 

340.  The  following  theorems  will  be  found  useful  in  determin- 
ing whether_/(a;j  •••a;,„)  is  continuous  at  a  or  not. 

From  277,  1  and  317,  1  we  have  at  once  : 

Letf(x-^---Xj,^^  g(x-^--'X„^  he  continuous  at  a.      Then 

f±g,  f-g^ 

f 

9 
are  continuous  at  a. 

341.  From  292  and  317,  2  we  have  at  once : 

Let  ^^^^^^(^^^...^j     ...     u„,  =  (fi,X^-^---x„,') 

be  continuous  at  x=  a=  (a^ •••«„).     At  x  =  a,  let 

u^  =  h^     ...     u,n  =  h„,. 
Let  /•/-  N 

be  continuous  at  u—b  =  (h^---  b,,^.      Then  y  considered  as  a  function 
of  the  x's  is  continuous  at  x  =  a. 

In  a  less  explicit  form,  we  may  state  this  theorem : 

A  continuous  function  of  a  continuous  function  is  a  continuous 

function. 

342.  In  order  that  f(x^"-x,n)  be  continuous  at  «,  it  is  necessary 
and  sufficient  that  for  each  e>0  exists  an  undeleted  vicinity  V(a), 
such  that  for  any  two  points  x\  x"  in  it., 

\f(x')-f(:x")\<e. 
This  follows  at  once  from  284  and  317,  1. 


210      CONTINUITY   AND   DISCONTINUITY   OF   FUNCTIONS 

Continuity  of  the  Elementary  Functions 

343.  The  integral  rational  fu7ictions  are  everywhere  continuous. 
^^^  y  =  x^.  n  positive  ititeyer. 

Then,  by  299,  y  is  continuous  at  every  point  x  in  9?.    Hence,  by 

340, 

UqX"  +  a^x"  1  + h  a„_^x  +  a„ 

is  everywhere  continuous.  Thus  the  theorem  is  proved  for  one 
variable. 

Let  y  =  'ZAxj^^x^^^  •  •  •  a;,/™, 

where  the  s's  are  positive  integers  or  0. 
Each  term  of  y,  viz. 

t  =  Ax-l^  ■  •  •  rc^*™,  CI 

is  continuous  at  an  arbitrary  point  x.     For, 

let  u^  =  x{\     •■■    Uj.^  =  Xjn'^. 

Then  the  term  t  becomes  the  product 

which,  considered  as  a  function  of  the  m's,  is  continuous,  by  340. 
On  the  other  hand,  each  u^  is  a  continuous  function  of  the  a;'s. 
Hence,  by  341,  f,  considered  as  a  function  of  the  a;'s,  is  continuous 
at  every  point  x  in  9^^.  Hence  y,  being  a  sum  of  the  terms  i,  is 
continuous,  by  340. 

344.  The  rational  functions  are  continuous  everywhere  in  their 
domain  of  definition  D. 

We  saw,  in  228,  that  the  domain  D  of 


y 


IBx^'^  •  •  •  Xjn'""       Gr 


embraces  all  points  of  9?„,  except  the  zeros  of  the  denominator, 
which  are  the  poles  of  y. 

By  343,  F  and  (r  are  everywhere  continuous ;  hence,  by  340,  y 
is  everywhere  continuous,  except  at  the  poles  of  y. 


DISCONTINUITY  211 

345.   1.    The  circular  functions  are  continuous  at  every  point  of 

their  domain  of  definition. 

From  202,  we  saw  that  sin  a;,  cos  a;,  are  defined  for  all  points 

of  9? ;  while 

sma;  ,         cos  a; 

tan  X  = ,  cot  X  =  —. — , 

cos  X  sin  X 

1  1 

sec  X  = ,  cosec  x  = 


cos  X  sin  X 

are  defined  for  all  points  of  9"?,  except  for  the  zeros  of  the  denomi- 
nators in  the  above  equations. 

From  296,  sin  x  and  cos  x  are  everywhere  continuous.     The  rest 
of  the  theorem  follows  now  from  340. 

2.   The  one-valued  functions 

arc  sin  a:,    arc  cos  x,   arc  tg  x,    arc  ctg  x 

are  continuous  at  every  'point  of  their  domains  of  definition. 
This  follows  at  once  from  294. 

346.  1.   The  exponential  functions  are  everywhere  continuous. 
This  follows  at  once  from  298,  5. 

2.    The  logarithmic  functions  are  everywhere  continuous  in  their 
domain  of  definition. 

Let  1 

y  =  \ogi,x. 

Then 

log^a; 

Hence  y  is  continuous  at  a  point  »,  if  log^a;  is.    But  this  is  con- 
tinuous for  every  a;>0,  by  300. 

The  demonstration  also  may  be  given  by  294. 

Discontinuity 

347.  If  ,.      ..  . 

lim/(a;i  •'•  x^) 

does  not  exist ;  or  if  it  exists,  and  is  different  from  /(a),  should 
f  be  defined  at  a,  we  say  /  is  discontinuous  at  a,  and  «  is  a  point 
of  discontinuity  of  f. 


212      CONTINUITY   AND   DISCONTINUITY   OF   FUNCTIONS 

Discontinuities  are  of  two  kinds  : 

Finite  discontinuities,  when  f  is  limited  in  F*(a). 

Infinite  discontinuities,  when  f  is  unlimited  in  every  F'*(a). 

348.  We  consider  now  in  detail  some  of  the  ways  in  which  a 
function  of  a  single  variable  /(a;)  may  be  discontinuous  at  a 
point  a. 

Finite  Discontinuities 
1.  /(a  +  0)=/(a-0)^/(a). 

Such  a  discontinuity  is  called  a  removable  discontinuity. 

Such  a  function  is 

y  =  lim  — 

considered  in  331. 


n=co  1    +  71X 


2.  /(a  +  0),/(a  —  0)  exist,  but  are  different. 

Such  a  function  is 

y  —  sgn  X, 
considered  in  330. 

3.  If  fi^x)  is  defined  at  a,  and  /(«)  =f{a  +  0),  we  say  /  is  con- 
tinuous on  the  right,  at  a. 

If /(a)  =f(a  —  0),/  is  continuous  on  the  left,  at  a. 

4.  Either /(a  +  0),  or/(rt  —  0),  or  both  do  not  exist. 

Such  a  function  is  ^ 

V  =  sin  -• 
^  X 

We  considered  this  function  in  256.     Here  neither  /(O  +  0)  nor  /(O  —  0)  exist 
Also  /  is  not  defined  for  x  =  0. 


Infinite  Discontinuities 

349.    1.  As  X  approaches  a  from  either  side,  f(x^,  either  mono- 
tone increases  or  mono-  ?/ 
tone  decreases. 


Ex.  1. 

atO. 

Ex.  2. 

y  =  Y 

atO. 

1- 

INFINITE  DISCONTINUITIES 


213 


2.  As  X  approaches  a,  /(a;)  increases  monotone  on  one  side,  and 
decreases  monotone  on  the  other. 


Ex.  3. 

atO. 
Ex.  4. 

at'!:. 


y  = 


y  =  tan  z, 


3.   As  X  approaches  a,  y  oscillates 
infinitely  often   about   a   base  curve, 
belonging  to  the  types  defined  in  1  or  2.     The  amplitude  of  the 
oscillations  is  limited. 


Ex.  5. 

Here  y  oscillates  about  the  base  curve 


y  =  — ha;  sin  -,   at  x  =  0. 

Xr  X 


and  the  amplitude  of  the  oscillations  converges  to  0  as  x  approaches  0. 


Ex.  6. 

at  X  =  0. 

Here  y  oscillates  about 


1  .  1 
y  =  -  +  sin-, 
^      X  x' 


y=-;  (1 

X  ^ 

and  the  amplitude  of  the  oscillations  remains  the  same,  viz.  ±1  above  the  curve  1). 

4,  The  discontinuities  considered  in  the  preceding  three  cases 

are  such  that  either  ,. 

lim  y 

x=a 

is  infinite,  or  at  least  the  right  and  left  hand  limits  at  a  are  infinite 
and  of  opposite  signs.  Such  points  of  discontinuities  of  fQc)  are 
called  infinities  ;  we  also  say /(a;)  is  infinite  at  such  points. 

5.  In  either  or  both  the  right  and  left 
hand  vicinities  of  a,  y  is  unlimited,  while  the 
corresponding  (infinite)  limits  do  not  exist. 


Ex.  7. 


1 


1 


w  =  -  sin  — 
^       X         X 


:0. 


Here  y  oscillates  between  the  two  hyperbolas 

1 
^  =  ±  X 

The  amplitude  of  the  oscillations  increases  indefi- 
Ditely  as  x  approaches  0. 


214      CONTINUITY  AND   DISCONTINUITY  OF   FUNCTIONS 

Ex.  8.  y  =  --}■  -  sin  —        a;  =  0. 

Here  y  oscillates  about  the  base  curve 

1 

y=  — 

X 

The  amplitude  of  the  oscillations  increases  indefinitely  as  x  approaches  0. 

1 
Ex.  9.  2/  =  e^. 

Here 

L lim y  =  0;    BMmy  =+  <x>, 

I  1 

Ex.  10.  y  —  6'  sin  — 

Here  L  lim  y  =  0  ;    i?  lim  y  does  not  exist. 

1=0  x=a 


Some  Properties  of  Continuous  Functions 

350.  1.  If  f(x^"-x^  is  continuous  in  a  limited  perfect  domain 
jD,  it  is  limited  in  D. 

For  if  /  were  not  limited, 

Max  I/I  =  +  00.  (1 

Then,  by  269,  there  is  a  point  a  of  i)  in  whose  vicinity  1)  holds. 
This  is  impossible.     For,  since /is  continuous, 

/(«i  •  •  •  O  -  e  </(a^i  •  •  •  2^m)  </(«!  •  •  •  a»)  +  e 
in  V(a). 

2.  The  theorem  1  does  not  need  to  be  true  if  I)  is  not  limited. 
Example.  -D  =  (0,  +00),  /(x)  =  x\ 

Here  /  is  continuous  in  Z),  but  /  is  not  limited. 

3.  The  theorem  1  does  not  need  to  hold  if  I)  is  not  perfect. 

Example.  D  =  (0*,  1) ,  /(x)  =  -• 

Here  /  is  continuous  in  D,  but  /  is  not  limited. 

351.  \.  At  x=a  let  f(x^---Xj^  he  continuous  and  =^0.     Then 

in  F(a), 

sgn  fi^i  "•^m)=  sgn /(«!  ---aj. 


SOME   PROPERTIES   OF   CONTINUOUS   FUNCTIONS         215 
For,  since  /  is  continuous  at  a,  we  have 

e>0,    8>0,    |/(a;) -/(«)!<€,  TsCa). 

Hence 

/(«)-€</(:r)</(a)  +  6. 

Since  e  is  arbitrarily  small,  we  can  take  it  so  small  that 

/(«),   /(«)-e,    /(a)  +  e 
all  have  the  same  sign. 

2.  The  theorem  1  gives  us  : 

At  x=  a  let  f(x^  - •  •  x,,^  he  continuous  and  4^ 0.     Then  there  exists 

a  p>0,  such  that 
^  |/|>p,         inF-(a). 

352.    Let /(a;^  •  •  •  a;„j)  be  defined  over  a  domain  2>.     By  defini- 
tion, it  is  continuous  in  D  when,  for  each  proper  limiting  point  x 

lim  f(x^  +  Aj  •  •  •  rc,„  +  h„,')  =  f{x^  •  •  •  a;„), 

ft=0 

the  points  x  +  h  lying  in  D. 

If  f{x-^^  +  Aj  •  •  •  x,,^  +  7i,„)  not  only  converges  to  f(^x^  •  •  •  a;^)  in  D, 
but  converges  uniformly^  we  say /is  uniformly  continuous  in  D, 

We  have  now  the  very  important  theorem  : 

If  f(x-^  •"X,n)  is  continuous  in  a  limited  perfect  domain  D,  it  is 
uniformly  continuous  in  D. 

Making  use  of  the  notation  of  325,  we  have  only  to  show  that 

\  =  Min  8,         for  I) 
is  >0. 

Suppose  it  were  not,  i.e.  let  3^  =  0.     We  show  that  this  assump- 
tion leads  to  a  contradiction. 

For,  by  269,  there  is  a  point  a  in  i),  such  that  in  V{a) 

MinS=So=0.  (1 

This  is  impossible.     In  fact,  by  342,  there  exists  for  each  e>0, 
a  8',  such  that  for  any  pair  of  points  x\  x"  in  V^-icC) 

\f(x'^-f(x"-)\<e.  (2 


216      CONTINUITY  AND  DISCONTINUITY  OF  FUNCTIONS 


Let  B"  <^8'.  Let  now  x'  be  any  point 
in  Vs-(a)- 

Then  every  point  x"  in  V&Ax'^  falls  in 
FaCa),  by  249. 

Thus,  for  any  such  pair  of  points  x\  a;", 
2)  holds. 

Thus,  for  no  point  x'  in  T'^--(a)  does  h 
sink  below  h" ,  and  this  contradicts  1). 


353.    Let  f(x-^  '"^m)  ^^  continuous  in  the  limited  perfect  domain 

D.      For  each  e  >  0  there  exists  a  cubical  division  of  i>,  of  7iorm 

S  >  0,  such  that 

\f(ix'^-fix")\<e  (1 

for  any  pair  of  points  x\  x"  in  any  one  of  the  cells  A  into  which  D 
falls. 

For,  since /is  uniformly  continuous  in  7),  let  a->0  be  such  that 
1)  holds  for  any  point  x"  of  V^(x'^.  Let  now  the  norm  of  the 
cubical  division  be 

Suppose  x\  x"  were  a  pair  of  points  in  some  cell  A,  such  that 
1)  does  not  hold.     Since 

Dist(a;',  x"~)<BVm<a;  by  244,  8,  9, 

x"  lies  in  V^x').     But  then  1)  holds  for  x',  x" .     We  are  thus 
led  to  a  contradiction- 


354.    1.   Let   f(x-^-'-x„^    he    continuous   in    the    perfect    limited 
domain  D.      Then  f  takes  on  its  extreme  values  in  D. 

Before  giving  the  demonstration,  let  us  illustrate  tlie  content  of  this  theorem. 
Let 


Then,  for  Z), 


J9  =  (0*,  1*),  and  y-x^. 
Max  y  =  +  1 ,    Min  y  =  0. 


But  y  does  not  take  on  either  of  these  extremes  in  D.     This  is  due  to  the  fact 
that  B  is  not  perfect. 


SOME   PROPERTIES   OF   CONTINUOUS   FUNCTIONS 


217 


2.  Let 


-O  =  (0,  +Qo),  and  y  = 


1 


Max  ?/  =  1,    Min  ?/  =  0. 

Thus  ;/  takes  on  its  maximum,  viz.  at  x  =  0,  but  does  not  take  on  its  minimum. 
This  is  due  to  tlie  fact  that  D  is  not  limited. 


3.  Let 
and 


I>  =  (0,  2), 
y  =  lim 


»i=x)  X"-  +  1 


for 
for 


This  function  is  a  particular  case  of  that  in  333. 

For 

0<a;  <  1,    y  =  x; 

x=l,   y  =  i; 

l<x<2,    y  =  0. 

Min  y  =  0,    Max  =  1. 


Hence 


The  function  takes  on  its  minimum  value  in  D,  but  not  its  maximum. 
This  is  due  to  the  discontinuity  of  y  at  1. 

355.    1.   We  give  now  the  demonstration  of  354. 

Let  e  be  an  extreme  of  f(^x^  •••  a?,„).  Then,  by  269,  there  is  a 
point  a  in  D  such  that  e  is  an  extreme  of  /  in  every  I^(a). 

Thus,  taking  e  >  0  small  at  pleasure,  there  is  at  least  one  point 
x'  in  any  Fi(a),  such  that 


\f(x')-e\< 


2 


(1 


Since  /  is  continuous  at  a,  there  is  a  8  >  0,  such  that  for  any  x 

in  VsQa') 


In  2)  set  x  =  x',  and  add  to  1);  we  get 

■      |/(«)-e|<e. 
/(a)=e. 


(2 


Hence,  by  87,  5, 


2.   As  corollary  we  have  : 

Let  /(x^  '  •  •  Xjj^)  be   continuous   and    >  0  in   the  perfect  limited 

domain  D.     Then  ^^.     ^     ^  .     -r^ 

Min/>0,         inD. 


218      CONTINUITY   AND   DISCONTINUITY   OF   FUNCTIONS 

356.  In  the  interval  %  =  (a,  h')  let  f(x)  he  continuous.  Let  it 
have  opposite  signs  at  a  and  h.  Then  f  vanishes  for,  some  point  c 
within  51. 

Let  us  form  a  partition  (^,  jB)  with  the  points  of  21.  The  class 
A  is  formed  thus.     Not  only  shall 

sgn/(a;)=sgn/(a)  (1 

at  every  point  of  J.,  but  between  a  and  any  point  of  A  shall  1) 
hold.     In  B  we  throw  the  other  points  of  21. 

Let  c  generate  this  partition.  Then  in  any  V(c^^f  has  opposite 
signs.  But  if  f{c)  were  =^  0,  by  351,  we  could  take  S  so  small 
that  f(x)  has  only  one  sign  in  Fg.  This  leads  to  a  contradiction. 
Hence  /(c)  =  0. 

The  point  c  cannot  be  an  end  point  of  21,  for  at  these  points 
/  T^  0  by  hypothesis. 

357.  Let  f(x)  he  continuous  in  21=  (a,  6).  Let  Minf(^x}=a^ 
Maxf(x)  =  ^  in  21.  Then  f(x)  takes  on  every  value  in  (a,  /3)  at 
least  once.,  while  x  passes  from  a  to  h. 

By  354,  f(x)  takes  on  its  extreme  values  in  21.     Let,  therefore, 
/(:r')=«,   f(x")=^. 

To  fix  the  ideas,  let  0  <  a  <  /3. 

Let  a<7</3.     Set 

9(x)=-f(x)-r^. 

Then  ,  ,.      „ 

ff<ix"}>0. 

Hence,  by  356,  g  vanishes  at  some  point  in  (a;',  a;").  At  this 
point  /(a;)  =  7. 

358.  Let  y  =  f(x)  he  a  continuous  univariant  function  in  the  inter- 
val (a,  5).  Let  a=f(a),  ^=f(Jiy.  Then  the  inverse  function 
x  =  g(^y^  is  a  one-valued  univariant  continuous  function  in  (a,  /3). 

By  214,  g(^y')  is  a  one-valued  univariant  function  in  its  domain 
of  definition  E.  By  357,  U  =  (a,  /3).  By  294,  g^y)  is  continuous 
in  (a,  /3). 


THE  BRANCHES  OF  MANY-VALUED   FUNCTIONS 


219 


The  Branches  of  Many-valued  Functions 

359.  Let  F(x^  •••  x,^  be  a  many-valued  function  in  D.  We  can 
form  a  one- valued  f unction /(a^j  •••  a;^)  over  a  domain  A<D  by 
assigning  to  /  at  each  point  of  A  one  of  the  values  of  F  at  this 
point,  according  to  some  law. 

A  common  way  to  do  this  is  to  assign  to /such  values  that  it  is 
continuous  in  A.  In  this  case  we  say  /(a;^  •••  x,n)  is  a  branch  of  the 
many-valued  function  F. 

A  point,  at  which  two  or  more  branches  meet,  may  be  called  a 
branch  point. 


360.   Ex.  1.    The  equation 


(1 


defines  a  two-valued  function  of  x  in  the  interval  (0,  +«)). 
One  branch  is  the  one-valued  function 


the  other  branch  is 


Vz; 
-Vx. 


(2 
(3 


The  graph  of  2)  embraces  the  points  in  the  upper  half 
of  the  parabola  1)  ;  the  graph  of  3)  is  the  lower  half  of 
this  parabola. 


361.   Ex.  2.  The  equation 

X*  —  ax^y  +  6?/3  =  0 

defines  a  three-valued  function  of  x  whose  graph  is  given  in  Fig.  1. 


Fig.  1. 

Still  preserving  the  continuity,  we  can  define  the  branches  of  y  in  several  ways, 
according  to  the  path  we  take  on  leaving  0. 


220      CONTINUITY   AND   DISCONTINUITY  OF   FUNCTIONS 

In  any  case,  however,  we  must  stop  at  the  points  A,  B,  at  which  the  ordinate  is 
tangent  to  the  curve.  For,  if  we  passed  beyond  these  points,  we  would  no  longer 
have  a  one-valued  function. 

In  Figs.  2,  3,  4  we  illustrate  various  ways  of  choosing  a  branch. 


Fig.  4. 


Fig.  2. 


Fig.  3. 


Notion  of  a  Curve 

362.  1.  In  elementary  mathematics  one  meets  with  a  great 
variety  of  curves.  Their  equations  may  be  expressed,  confining 
ourselves  for  the  moment  to  the  plane,  in  one  of  the  three  forms 


F(xy')=0. 


(1 
(2 
(3 


When  the  curve  is  given  by  1)  or  2),  it  is  said  to  be  defined 
explicitly  ;  when  given  by  3),  it  is  said  to  be  defined  implicitly. 

We  observe  that  1)  is  a  special  case  of  2).     For  we  have  only 
to  write 

X  =  U,    y  =  ylr{u), 

to  reduce  1)  to  2). 

When  the  curve  is  given  by  2),  it  is  said  to  be  expressed  in  para- 
metric form. 

We  note  that  1)  is  also  a  special  case  of  S).     For  we  have  only 
to  set 

F(xy^=y-f{x\ 

to  bring  1)  to  the  form  3). 


NOTION   OF  A  CURVE  221 

2.  It  is  customarily  thought  that  the  notion  of  a  curve  is  a 
very  simple  one ;  but  we  shall  see  that  this  is  not  so.  On  the 
contrary,  it  is  a  very  obscure  and  complex  notion.  Reserving  the 
discussion  of  the  notion  of  a  curve  in  general  until  later,  it  is  well 
to  give  a  preliminary  definition. 

Let  <^(m),  T/r(w)  be  continuous  one-valued  functions  of  u  for  an 
interval  51  =  («,  h^t  finite  or  infinite. 
Set 

Let  u  range  over  %.  The  points  P  whose  coordinates  are  x,  y 
will  form  a  point  aggregate  which  we  call  a  curve.  The  point  x,  y 
is  the  image  of  the  point  u. 

Ex.  1,  Let  (p(u)=  u,  \p{u)=  tfi.  The  curve  so  defined  is  a  parabola  whose 
axis  is  the  ^/-axis. 

Ex.  2.     Let  0  (?()  =  a  cos  ?<.,  i/*  (?<)  =  h  sin  ?j.     The  curve  so  defined  is  an  ellipse. 

Still  more  generally  an  aggregate  of  a  finite  number  of  curves 
may  be  called  a  curve  0. 

Each  one  of  the  individual  curves  which  enter  in  O  may  be 
called  an  arc  ov  part  of  C 

If  a  curve  or  a  piece  of  a  curve  is  such  that  ?/  is  a  one-valued 
monotone  function  of  x,  we  shall  say  the  curve  or  the  piece  of  it  is 
monotone.  In  the  same  way  we  shall  extend  the  terms  monotone, 
increasing,  univariant,  etc.,  to  curves  or  arcs  of  curves. 

3.  Let  u  range  from  a  to  h.  If  to  two  different  points  w',  u" 
of  21  =  (a,  &)  corresponds  the  same  point  P(x,  «/),  this  point  P 
will  be  called  a  multiple  point  of  0. 

Let  the  coordinates  xy  take  on  the  same  pair  of  values  at  the  end 
points  of  51  =  (a,  J).  Then  C  is  called  a  closed  curve.  If  O  has 
no  multiple  points  in  (a,  5*),  we  shall  say  (7  is  a  closed  curve  with- 
out multiple  points. 

4.  The  extension  of  these  notions  to  w-dimensional  space,  by 
setting  ,    ^  N  ,   ^  X 

is  too  obvious  to  need  comment. 


CHAPTER  VIII 

DIFFERENTIATION 

FUNCTIONS  OF  ONE   VARIABLE 

Definitions 

363.   Let  y=f(x)  be  defined  over  a  domain  D  for  which  «  is  a 
proper  limiting  point.     The  quotient 

Aa;  X  —  a 

is  called  the  difference  quotient  at  a. 
If  we  set  2;  =  a  +  A,  we  have  also 


Ay_/(a  +  A)-/(a) 
Aa;  Ti 


(2 


lim^  =  77  (3 

exist,  finite  or  infinite.     Then  t]  is  called  ^Ae  differential  coefficient 
of  /(a;)  at  a,  and  is  denoted  by 

Let  A  be  the  aggregate  of  points  in  D  for  which  77  is  finite  or 
infinite.  The  corresponding  values  of  77  define  a  function  of  x^ 
called  the  derivative  of  fix)^  more  specifically,  the  jirst  derivative 
of  fQc).     It  is  represented  variously  by 

/'(.).   BJi.^,   |,   |.  (4 

222 


GEOMETRIC   INTERPRETATIONS 


223 


The  function  /'(a;)  is  said  to  be  obtained  from  f(x)  by  the 
process  of  differentiation.  A  function  which  admits  a  derivative 
is  said  to  be  differentiable. 

Since  /'(re)  may  be  infinite,  the  reader  will  observe  that  its 
values  lie  in  M-     Cf.  276. 

364.   In  the  same  way,  the  right  and  left  hand  limits  at  x  =  a, 


i^lim 


A?/ 


L  lim 


Ay 

A^' 


give  rise  to  right  and  left  hand  differential  coefficients  at  a.     These 

we  denote  by  t-p'^  \ 

i^/'(a),     Z/'(a). 

These  in  turn  give  rise  to  right  and  left  hand  derivatives,  which 
we  may  denote,  prefixing  R  and  L  before  the  symbols,  4)  in  363. 

When  speaking  of  differential  coefficients  and  derivatives  in  the 
future,  we  shall  mean  those  defined  in  363,  unless  the  contrary  is 
expressly  stated. 

However,  much  that  we  prove  for  /'(a)  and  /'(a:)  may  be 
applied  at  once  to  the  corresponding  unilateral  differential  coeffi- 
cients and  derivatives. 


Geometric  Interpretations 
365.   Let  P  and  R  be  the  points  on  the  graph  of 

1/  =/(a^). 

corresponding  to  x  =  a  and  x  =  a  +  Ax^ 
Fig.  1. 
Then 

PW=Ax=h,   BW=At/. 

If  the  secant  PR  makes  the  angle 
<f>  with  the  a;-axis,  ^  _,  .^ 


Fig.  1 


Ay 
Ax~PW 


=  tan  <^. 


That  is :  the  difference  quotient  is  the  tangent  of  the  angle 
secant  makes  with  the  x-axis. 


that  the 


224 


DIFFERENTIATION 


Suppose  now  y  is  continuous  in  a  little  interval  about  x  =  a\  if 
the  secant  PR  approaches  a  limiting  position  P  U^  as  R  approaches 
the  fixed  point  P  from  either  side,  we  say  PU  is  the  tangent  to  the 
curve  at  P. 

Evidently,  if  /'(«)  is  finite. 

Ay 
f'(^a}  =  lim  -r^  =  lim  tan  cj)  =  tan  a, 

where  a  is  the  angle  that  the  tangent  line  makes  with  the  a;-axis. 
If  f'(a')=  ±00,  the  tangent  line  is  parallel  to  the  ?/-axis. 


/(«;=+ 00 


Y 


Fig.  2. 


.TT 


P 

/'(CO  =-00 


Fig.  3. 


Such  cases  are  shown  in  Figs.  2  and  3. 

The  point  ^  is  a  point  of  inflection  with  vertical  tangent. 

For  an  example  of  such  a  function,  see  388,  5. 


366.  1.  When  the  differential  coefficient  at  a  does  not  exist, 
finite  or  infinite,  the  right  and  left  differential  coefficients  may. 
They  are  then  different. 

If  both  are  finite,  we  have  a  case  illustrated  by  Fig.  1. 


Sucli  a  function  is 


^/  N        e^  -  1 


for  x  9^  0  ; 


Here 


e^+  1 
=  0,         for  X  =  0. 
i2/(0)  =  +l,   i/'(0)  =  -l. 


If  one  is  finite  and  the  other  infinite,  we  have 
a  case  illustrated  by  Fig.  2. 

The  points  P  in  Figs.  1,  2  are  called  angular 
points. 


NON-EXISTENCE   OF   DIFFERENTIAL   COEFFICIEN'J'      225 

2.  When   both  differential  coefficients  are  infinite,  but  of  op- 
posite signs,  we  have  a  case  illustrated  by  Figs.  3,  4. 


Lf{u)  =  - CO 
R/'(a)  =  -t-m 

Fig.  3. 


Lf  (,a)  =  -i- CO 
B/'(a)  =  -co 

Fig.  4. 


Here  I*  is  a  cusp  with  vertical  tangent. 

See  388,  3  for  an  example  of  sucli  a  function. 

3.  In  Case  1  the  curve  has  not  one  but  two  tangents  at  P ;  viz. 
a  right  and  a  left  hand  tangent.  Case  2  may  be  considered  as  a 
special  or  limiting  case  of  1.     The  curve  has  a  tangent  at  P. 

In  both  cases  the  direction  of  motion  along  the  curve  changes 
abruptly. 

When  we  say  "a  curve  has  at  every  point  a  tangent,"  we 
exclude  Case  1. 


Non-existence  of  the  Differential  Coefficient 

367.  1.  We  consider  now  some  examples  of  continuous  func- 
tions for  which  the  differential  coefficient  on  either  side  of  certain 
points  does  not  exist. 

Let 


=  0, 


TT 


for  x^Q 


X 

for  a;  =  0. 


The  graph  F  of  ?/  is  given  in  the  adjoining 

figure.  Q 

Evidently    F   oscillates   between   the    two 

lines 

y=±x,  (1 

with  increasing  rapidity  as  x  approaches  0.. 


226  DIFFERENTIATION 

For  a:=5tO,  ^  is  evidently  continuous. 
For  a;  =  0,  y  is  also  continuous,  since 

lim  X  sin  —  =  0. 

1=0  X 

At  the  origin  the  secant  line  OP  oscillates  between  the  two 
lines  1),  and  obviously  does  not  approach  any  fixed  position  as  P 
approaches  0  from  either  side.     Thus  F  has  no  tangent  at  all  at  0. 

This  result  is  verified  at  once  analytically. 

For, 

^y  .       IT  .  n 

—^  =  sm  -i— 1          at  a;  =  U ; 

Ax  Ax 

and  as  Ax  =  0,  sin  -r—  oscillates  infinitely  often  between  ±  1. 

dy  1 

2.   For  use  later,  let  us  find  ~  for  a;  =  — 

ax  n 

We  have,  setting  Ax  =  A, 


¥^l('-+k\ 


+  h  ]  sm 


Ax  h\n  J  1  , 
-  +  h 
n 


1  +  wA    .        nir 

sm 


But 

sin 


-  =  sm   TiTT  — =— (— l)"sm- 

1  +  nh  \  1  +  nhJ  1  +  nh 

Hence,  setting 


u  — 


1  +  nh 


and  thus 


^y  ^     ix«       sinw 

Ax  u 

T     ^V         ^     -«N„      T     sinM 
lim  — ^  =  —  (  —  lynir  lim 

=  -(-l)"W7r,  by  301. 


NON-EXISTENCE   OF   DIFFERENTIAL   COEFFICIENT      227 


368.    Let       y  =f(x)  =  x^  sin—,  a;  ^t  0  ; 

X 


=  0, 


x  =  Q. 


Evidently  y  is  everywhere  continuous 
even  at  0. 

Tlie  graph  T  oi  y  oscillates  between  the 

two  parabolas 

y  =  ±x^ 

with  increasing  rapidity  as  x  approaches  0. 

As  P  approaches  0,  the  secant  OP  oscillates  between  narrower 
and  narrower  limits,  which  limits  converge  on  both  sides  toward 
the  2;-axis.     Evidently, 

/(0)  =  lim^=0; 

and  r  has  a  tangent  at  0,  viz.  the  axis  of  x. 
This  result  is  verified  analytically  at  once. 
For, 


and 


-r^  =  A3;sin— -  at  0, 
Aa;  Aa: 


lira  Aa;  sin-^— =  0. 

Ai=o  Aa; 


369.   Let  A  =  0,  ±1,  ±  |,  ±\,  ... 

For  X  not  in  A,  let  y  =f(x~)  =  x sin—  sin- 

sin  — 

X 

For  X  in  A,  let  y  =  0. 

Here  y  is  everywhere  continuous,  even  at  the  points  of  A. 

Let  C  be  the  graph  of  y^  and  V  the  graph  of 


y^  =  x^\n-, 

considered  in  367. 

In  Fig.  1,  the  full  curve  represents  an  arc 

of    r    for    an    interval    /„  =  (a„,  5„),    «„  =  -, 

1  ^ 

6„= -•      The  dotted  curve,  call  it  P,  is 

n  —  \ 

symmetrical  to  V. 


Fig.  1 


228 


DIFFERENTIATION 


We  observe  now  that  y  is  obtained  by  multiplying  the  ordinate 
y-^  of  r  by  the  factor 


2/2  =  sin - 


Sin— 

X 


As  X  approaches  an  end  point  of  i^. 


sin  —  =  0. 


Hence  y^^  oscillates  infinitely  often  between  ±  1.     The  effect  of 
the  factor  y^  in  y  =  y-^y^  is  thus  to  bend  F  in  I„  an 
infinite  number  of  times,  so  that  the  resulting  curve, 
a  portion  of  C,  lies  between  F  and  F'. 

This  is  represented  in  Fig.  2,  where  the  light  and 
dotted  curves  are  F  and  F',  and  the  heavy  curve  is  C. 

At  one  of  the  points  of  A,  as  a„,  the  secant  a^P 
oscillates  with  increasing  rapidity  as  P  approaches 
a„  from  either  side. 

Since 

lyrnr,  by  367,  2, 


dx 


Fig.  2. 


the  tangents  to  F  and  F'  are  not  the  a:;-axis.  Hence 
the  limits  of  oscillation  of  the  secant  do  not  converge 
to  0,  and  hence  the  secant  a„P  does  not  converge 
to  some  fixed  position  as  x  approaches  a„. 

Thus  y  has  no  differential  coefficient  at  any  point  of  A,  and  its 
graph  O  has  at  these  points  no  tangent. 

Since  0  is  the  limiting  point  of  A,  there  are  an  infinity  of  these 
singular  points  in  the  vicinity  of  the  origin. 


370.    Let  ^  =  0,    ±1,   ±2, 


TT 


TT 


For  X  not  in  A,  let  y  =  f(x)  =  x^  sin  —  sin 
For  X  in  A,  let  ^  =  0.  ^"^  ^ 

The  reasoning  of  369  may  be  applied  here.     The  graph  of  y 
oscillates  between  the  two  curves 


discussed  in  368. 


y-.  =  ±  x^  sm  — , 
^  X 


FUNDAMENTAL   FORMULA   OF   DIFFERENTIATION       229 

There  is  no  tangent  at  the  points  ±1,  ±2,  •  •  •  while  at  the  origin 
there  is  a  tangent,  viz.  the  a;-axis. 

The  graph  O  oi  y  presents  therefore  this  peculiarity :  in  the 
vicinity  of  the  origin  there  are  an  infinity  of  points  at  which  0 
has  no  tangents ;  yet  at  the  origin  itself  C  has  a  tangent. 

371.  In  369  and  370,  the  aggregate  A  is  of  the  first  order,  by 
263,  2. 

It  is  easy  by  the  process  of  iteration  to  form  continuous  func- 
tions which  have  no  differential  coefficient  over  an  aggregate  A, 
of  order  m. 

Let  6(x)  =  sin  — 

and 

y  =  xd(x)d'-\x)  •••  e^^+^\x^. 

This  expression  does  not  define  y  at  points  involving  division 
by  zero.  At  these  points,  call  their  aggregate  A,  we  set  y  =  0. 
It  is  easy  to  show  that  y  is  everywhere  continuous  and  that  it  has 
neither  right  nor  left  hand  differential  coefficients  at  any  point  of 
A,     The  aggregate  is  of  order  m.     See  259,  260.  • 

Fundamental  Formulce  of  Differentiation 

372.  As  many  American  and  English  works  on  the  calculus 
derive  these  formula?  in  an  incorrect  or  incomplete  manner,  we 
shall  deduce  some  of  them  here.  We  shall,  at  the  same  time, 
prove  them  under  conditions  slightly  more  general  than  usual. 
As  domain  of  definition  D  of  our  functions  y,  u^  v,  -••  we  take 
any  aggregate  having  proper  limiting  points.  The  domain  of 
definition  A  of  their  derivatives  will  embrace,  at  most,  the  proper 
limiting  points  of  D. 

It  is  convenient  to  represent 

y(x-\-]i)^  u(x+K)^   ••'  by  y,  u,  •••  etCo, 

-1  dy    du  -I        ,      f  . 

and.  -^,  — ,  •••  by  y,  w,   •••  etc. 

dx    dx 


230  DIFFERENTIATION 

373.  We  begin  by  proving  : 

If  the  differential  coefficient  f  (^a)  is  finite^  f{x)  is  continuous  at  a. 
For,  since 

A=o  h 

we  have,  for  each  e >  0,  a  S  >  0,  such  that,  if  |  A|  <  S, 

f(a  +  A)  —f(a')      j?i  r  \  ,    I  I  M  ^ 

n 
Hence 

f(a  +  A)  =/(a)  +  A(/'(a)  +  e'). 
Therefore, 

lim/(a  +  A)  =fCa}, 

which  states  that  /  is  continuous  at  a. 

374.  //^  ?/  ^s  constant  in  D,  i/'  =  0. 
For 


for  any  point  of  D. 


Ax 


375.  ie^  y  =  u±v.     Let  u' ^  v'  be  finite  in  A.     Then  y'  =  u'  ±  v' 
in  A. 

For  A^  _  Aw     Ay  ^-. 

Ax      Ax     Ax 

Since  w',  v'  exist  and  are  finite,  we  can  apply  277,  2,  to  1). 

376.  Let  y  =  uv.     Let  w',  y'  he  finite  in  A. 
Then,  in  A, 

y'  =  wv'  +  vu' .  (1 

For  _ 

Ay  _uv  —  uv  _  (u  +  Am)  (^  +  A'^)  —  uv 

Ax         Ax  Ax 

uAv  -\-  vAu      -Av  ,     Au  ^o 

= ■ =  u \-v — -•  {z 

Ax  Ax        Ax 


FUNDAMENTAL   FORMULA   OF  DIFFERENTIATION       231 

By  373,  ..     _• 

lim  u  =  u. 

By  hypothesis, 

lim  — ^  =  %',  lim— =  t/. 

Aa;  Ax 

Hence,  passing  to  the  limit  in  2),  we  get  1). 

377.    1.  Let  y=-'     Let  u',  v'  be  finite  and  v=^0,  in  A.     Then 

f      vu'  —  uv'      .     A  /  -1 


(2 


For  Ay  _  vAu  —  uAv  _  1  Aw  _  w    1  Av^ 

Ax  vvAx  V  Ax      V    V  Ax 

By  373, 

lim  V  =  v. 

By  hypothesis, 

T     Aw        ,    T     Av        f 
nm — =u',  nm  —  =  v'. 

Ax  Ax 

Passing  now  to  the  limit  in  2),  we  get  1). 

We  observe,  by  351,  that  v=^0  for  Ax  sufficiently  small,  since  v 
is  continuous  and  r^t  0  at  a;.  It  is  therefore  permissible  to  divide 
by  V,  as  in  2). 

2.    Criticism.     Some  writers  derive  1)  as  follows.     From 

u 

they  get 

yv  =  u. 

They  now  apply  376,  which  gives 

u'  =  yv'  +  vy'y  (3 

which,  solved,  gives  1). 

This  method  is  incorrect.  For  to  get  3),  by  using  376,  we 
must  impose  the  condition  that  y'  exists  and  is  finite.  But  noth- 
ing in  this  form  of  demonstration  shows  the  existence  of  y' .  The 
method  then  shows  only  this :  on  the  assumption  that  y'  exists, 
its  value  is  given  by  1).  But  this  assumption  of  existence  makes 
the  demonstration  worthless. 


232  DIFFERENTIATION 

3.  Many  writers  of  elementar}^  mathematical  text-books  are  not 
alive  to  the  fact  that  a  demonstration,  which  involves  an  assump- 
tion of  the  existence  of  certain  quantities  or  forms,  renders  the 
demonstration  invalid.  This  error  of  reasoning  is  extremely  com- 
mon in  the  calculus.  Because  determinate  results  are  obtained  by 
such  reasoning,  it  is  allowed  to  pass  as  conclusive. 

To  show  how  fallacious  this  style  of  reasoning  is,  let  us  assume 
that  we  can  write  * 

1 


c2-4 


=  a  sin  X  -\-h  cos  x. 


Granting  this,  it  is  easy  to  determine  a  and  h.     In  fact,  setting 
a;  =  0,  we  get 

Setting  a;  =  — ,  we  get 

4 
a  = 


7r2  -  16 
Hence 


;  sm  X cos  X, 


a;2  _  4      ^2  _  16  4 

a  perfectly  determinate  result;  but  also  a  perfectly  false  result. 
In  fact,  the  right  side  of  4)  is  a  periodic  function,  while  the  left 
side  is  not. 

The  reader  should  therefore  not  begrudge  the  pains  it  is  some- 
times necessary  to  take,  to  prove  an  existence  theorem.  He  should 
also  notice  that  by  modifying  the  form  of  proof  it  is  sometimes 
possible  to  avoid  assuming  the  existence  of  certain  things  which 
enter  the  demonstration.  Witness  the  demonstrations  just  given 
of  1)  in  1,  2. 

378.    Let  y=f(x'),  and  x  =  g(t).     Let  g'(t^  =  —  he  finite  in  T. 

n  dt 

Let  X  he  the  image  of  T.     If  -^=f'  (a;)  is  finite  in  X, 

dy^  _dy^    dx  ^1 

dt       dx    dt 

*  In  treating  the  decomposition  of  a  rational  function  into  partial  fractions,  it  is  often 
assumed,  ivithout  any  jitstification,  that  the  decomposition  in  tlie  form  desired  is  possible. 


FUNDAMENTAL    FORMULA   OF   DIFFERENTIATION       233 

Before  proving  this  theorem,  we  wish  to  illustrate  two  cases 
which  may  occur. 

Ex.  1.     Let  x  =  t  sin  2  mirt.     The  period  of  sin  2  mtrt  considered  as  a  function  of 

t  is  —     By  taking  m  very  large  but  fixed,  x  will  oscillate  a  great  many  times  near 
m 

the  origin.     Where  the  graph  cuts  the  «-axis,  i.e.  when 

A.       ,     1         ,1,3 
2  m         m         2  m 
we  have  Ax  =  0. 

But  however  large  m  is  taken,  we  can  determine  a  5  >  0,  such  that  Ax  ^t  0,  in 

F5*(0).     In  fact,  we  have  only  to  take  5  < 

2  m 

What  we  have  shown  for  i  =  0  is  true  for  any  other  point  t.  That  is,  we  can 
always  choose  5  sufficiently  small  so  that  in  V^*(J,),  Ax  shall  not  =0. 

Ex.  2.  x-fi sin -,fovt^O; 

=  0,  for  t  =  0. 

The  graph  of  this  function  we  considered  in  368.  For  any  point  «  ^t  0  we  can 
determine  a  5  such  that  in  Vs*(,t),  Ax  does  not  vanish.  Not  so  at  <  =  0.  Here, 
however  small  5  >  0  is  taken,  x  oscillates  infinitely  often  in  Fg*(0) ;  and  thus  for  an 
infinity  of  points  in  F5*(0),  Ax  =  0. 

We  can,  however,  throw  the  points  of  ^^^(O)  in  two  sets.  In  one,  call  it  Vo,  we 
put  the  points  for  which  Ax  =  0.     Then 

Vo  =  ±-,   ± 


m        m  +  1 

In  the  other  set,  call'it  Fj,  we  put  all  the  other  points  of  V*.  We  can  now  show 
for  the  function  y  =/(x)  in  the  above  theorem,  that  1)  is  true  for  each  one  of  these 
sets  of  points,  and  therefore  true  for  both  together. 


379.    We  give  now  the  proof  of  378. 

Let  ^  be  any  point  in  T;  let  x  be  the  corresponding  point  in  X. 
Let  Aa;,  Ai/  be  the  increments  of  a;,  y,  corresponding  to  the  incre- 
ment At  of  t. 

Case  1.      There  exists  a  V*(^t^,  in  which  Ax=f=0. 
The  identity 

At      Ax'  At  ^ 

does  not  involve  a  division  by  0,  as  Ax  ^  0. 


234  DIFFERENTIATION 

Siiice  —  =  g'(t')  is  finite  at  t^  Aa;  =  0  when  Ai  =  0.     Hence,  by 
292,  ^* 

lim^  =  lim^  =  f. 
A«=o  Aa;       Ai=o  Aa;      aa; 

Thus, 

lim  -~  =  lim  -r-^  •  lim  -r— , 
A«=o  Ac        Ax=o  Aa;      Afc=o  Ac 

and 

c?^      dx     dt 
which  proves  the  theorem  for  this  case. 


(2 


Case  2.     Aa;  =  0  for  some  point  in  every  V*(f). 
Let  Vq  be  the  points  of  V*(t)^  for  which  Aa:  =  0. 
Let  V^  be  the  remaining  points  of  V*(f). 

If  we  show 

T     Ay  ,    dy  ^.     Aa;  ^o 

iim-r^,   and    ^lim-r— ,  (o 

At  cZa;         Ar  ^ 

have  one  and  the  same  value  for  every  sequence  of  points  whose 
limit  is  t,  we  have  proved  2)  for  this  case. 
Let  A  be  any  sequence  in  V^.     Then 

lim  — =  0,  (4 

since  Aa:  =  0  for  every  point  in  A. 


As  ~  is  finite  at  a;, 
dx 


On  the  other  hand, 


dy  ^.  Aa;  ^ 
-/  hm  -i—  =  0. 
c^a;    A    Ai 


lim^  =  0. 

^    A^ 


For,  Aa;  being  0  for  every  point  of  vl,  y=fQc)  receives  no 
increment,  and  hence  Ay  =0  in  ^. 

Thus,  for  every  sequence  A^  the  two  limits  in  3)  have  the  same 
value,  viz.  0. 


FUNDAMENTAL   FORMULA   OF   DIFFERENTIATION       235 

Let  now  B  be  any  sequence  in  F^  which  =t.     Let  the  image 
of  the  points  B  be  the  points  (7,  on  the  rc-axis. 
Then,  by  292, 

hm  — ^  =  hm  -r^  •  hm  -i— 

B       iid  C      I!^X  B      ^t 

=  ^lim^.  (5 

dx    B    ^t 

Thus  the  two  limits  of  3)  are  the  same  for  each  sequence  B. 
It  remains  to  show  that  one  is  0.     Now,  by  4), 


since,  by  hypothesis, 


lim  —  =  lim  —  =  0 ; 

B      M  A       M 


lim  — =  a'(0 


for  any  sequence  whose  limit  is  the  point  t. 

Hence  the  right  side  of  5)  is  0. 

Thus  the  two  limits  3)  have  the  value  0  for  every  sequence  A 
or  B.  These  limits  therefore  have  tlie  value  0  for  any  sequence, 
whether  its  points  all  lie  in  F^,  or  in  J\^  or  partly  in  Vq  and  partly 
in  Fj. 

380.  The    demonstration,    as    ordinarily   given,    rests    on   the 

identity 

•^  Ay  _Ai/     Ax 

'At  ~  Ax  '  At' 

The  theorem  is,  therefore,  only  established  for  functions  x=g(t)^ 
which  fall  under  Case  1. 

If  one  wishes  to  give  a  correct  but  elementary  demonstration, 
it  would  suffice  to  restrict  g(t)  to  have  only  a  finite  number  of 
oscillations  in  an  interval  T^  and  have  at  each  point  of  ^  a  finite 
differential  coefficient.  In  an  elementary  text-book  on  the  cal- 
culus it  is  not  advisable  to  consider  functions  with  an  infinite 
number  of  oscillations. 

381.  Let  y  =f(x)  he  univariant  and  continuous.  Let  x  =  g(y^  be 
its  inverse  function.  Let  f  (x)  he  finite  or  infinite  in  A.  Let  E  he 
the  image,  of  A.     Let  x  and  y  he  corresponding  points  in  A  and  E. 


236 


DIFFERENTIATION 


If  f  (x)  is  finite  and  ^  0,  then  g'  Qy^=  — 

f  (P) 

jf  f(x)={)  then  a'(v')=  {  +  °^  '^^  ''  increasing. 
^  '    ^  ^       '  ^  ^^^      [  -co  if  f  is  decreasing. 

If  f  (x)  is  infinite,  g'  (jy^=  0. 

Since  /  is  iinivariant,  Ay  and  therefore  also  -^  are  4^  0. 

Hence  the  relation  .  ^ 

A.r  _   1 

A^~A^ 

Aa; 


(1 


does  not  involve  for  any  point  a  division  by  0. 

Since  y  is  continuous,  Ay  =  0  when  Aa;  =  0. 

We  have  therefore  only  to  apply  292  in  passing  to  the  limit 
inl). 


382.    The  geometric  interpretation  of  381  is  very  simple  in  the 
following  case : 


a    x■^x  Xg  5 


Let  y  z=i  f(x)  be  a  continuous  increasing  function  in  (a,  5). 

The  inverse  function  x  =  g(jf)  is  increasing  and  continuous  in 
(a,  /3).     See  Fig. 

The  graph  of  f{x)  and  g(jf)  is  the  saqie  curve  G.  At  Pj,  P^ 
we  have  points  of  inflection. 

If  Pr  is  the  tangent  at  P, 


tan  </)=/'  (a;)  = 


tan^  =  /(y)  = 


dx 

dx 
dy 


FUNDAMENTAL   FORMULA   OF  DIFFERENTIATION       237 

Since 


e  +  cf>  = 

TT 

=  2' 

tan  0  = 

1 
tan 

4 

dx  _ 
dy 

1 

~  dy 
dx 

or 


The  consideration  of  the  tangents  at  Pj,  P^  illustrates  the 
theorem  for  the  other  cases. 

383.  We  apply  the  preceding  general  theorems  to  find  the 
derivatives  of  some  of  the  elementary  functions,  choosing  those 
whose  demonstration  is  often  given  incorrectly. 


i>x= 

;  w^  log  a.        a  >  0,  a;  arbitrary. 

(1 

For,  let 

y  =  a\ 

Then 

Aa:               Aa: 

(2 

But,  by  311, 

n^x  _  1 

lim  — =  log  a. 

Ax=fl      Aa;  ^ 

Passing  to  the  limit  in  2),  we  get  1). 
When  a  =  e,  1)  becomes 

D^e""  =  e^  (3 

384.    1.  i>^loga;  =  -.        x>0.  (1 

Let 

Then 

But 


From  2)  we  get,  by  381, 
which  is  1). 


>x  log  X  = 

1 

x' 

y  = 

■■  log  X. 

x  = 

■.e\ 

dx 
dy 

--e«  =  x. 

dx 

1 

X 

(2 


238  DIFFERENTIATION 

2.  We  can  get  1)  directly  as  follows : 

Ay  _  ]og(x  +  Ax')  —  log  X  _     ^\         x 


Ax 

1 

X 

Ax 
log(l+  J 

Ax 

But, 

by 

X 

310  and  292, 

log(l  + 

lim        ^  ^ 

Ax=o           Ax 

Ax 


Hence,  passing  to  the  limit  in  3),  we  get  1)  again. 
From  1)  we  can  prove  again 


D^e"^  =  ef. 

For,  from 

y==e\ 

we  have 

X  =  log  y. 

Hence,  by  1), 

dx  _1 

dy    y 

Using  381,  we 
which  is  5). 

have 

dy 

(3 


(4 


(5 


385.    1.   Criticism.     In  either  of  the  preceding  ways  of  getting 

B^e^  and  D^  log  a;, 
we  need  the  limit  j 

lim  (1  +  w)«  =  e.  (1 

u=0 

Some  writers  only  prove  1)  when  u  runs  over  the  sequence 

2^  ^'  4'  "• 

Others  prove  1)  only  for  a  right  hand  limit.     As,  however.  Ax 

may  have  any  positive  or  negative  values  as  it  converges  to  0,  the 

limit  1)  must  be  established  without  any  restriction. 


FUNDAMENTAL   FORMULA   OF   DIFFERENTIATION       239 

2.  If  the  method  of  384,  2  is  used  to  get  D^  log  a;,  we  must  not 
only  prove  1),  but  we  must  show  that 

1  1 

lim  log  (1  +  uy  =  log  lira  (1  +  m)«. 

u=0  „=0 

This  is  rarely  done. 

3.  A  third  method  is  to  employ  the  Binomial  Theorem,  which 
is  taken  from  algebra. 

The  rigorous  demonstration  of  this  theorem  for  any  case,  besides 
that  of  integral  positive  exponents,  is  far  beyond  the  limits  of  the 
ordinary  high  school  or  college  algebra.  Moreover,  the  demon- 
strations usually  given  are  incorrect.  The  employment  of  the 
Binomial  Theorem  to  find  the  above  derivatives  is  therefore  open 
to  the  most  serious  criticism. 

386.  1.  The  differentiation  of  the  direct  circular  functions  pre- 
sents nothing  of  note ;  let  us  therefore  turn  at  once  to  the  inverse 
circular  functions. 

We  take  .  ^^ 

y  =  arc  sin  x  (1 

as  an  example.     The  notation  indicates  that  we  have  taken  the 
principal  branch  of  arc  sin  a;,  [223].     Then 

-2<3'<2'  ^^ 

From  1)  we  have 

a:  =  sin  y. 

Hence  , 

ax 


,    =  cos  y  =  Vl  —  x^,  CS 

dy  ^ 

The  radical  has  the  positive  sign,  as  cos^  is  not  negative  for 
the  values  2). 
Hence,  by  381, 

-^  =  i)^  arc  sin  a;  =  —  •>        \x\^l. 

dx  Vl  -  a;2 

=  +  00  for  a:  =  ±  1. 

2.  Criticism.  In  many  books  the  branch  of  arc  sin  x  which  is 
taken  is  not  specified.  Consequently,  the  sign  of  the  radical  in 
3)  is  not  specified.  For  some  branches  the  negative  sign  should 
be  taken. 


240 

DIFFERENTIATION 

387.    1. 

D^- 

=  fxx'^- 

-^           a;  >  0,      fx  arbitrary. 

(1 

Let 

y  =  x>'. 

Then 

^^gA^loga;^ 

(3 

Let 

fl  log  X  =  M. 

Then 

y  =  e\ 

But 

dy  _dy  du  ^ 
dx      du  dx 

and 

• 
du            dx      X 

Hence 

dx 

2.    Criticism.     Some  writers  rest  the  demonstration  on 

log(l  +  w)  _ 


lim 

«=o  U 

and  are  thus  open  to  the  criticism  of  385,  2.     Others  proceed  thus. 

From  2)  we  have 

\ogy  =  fi\ogx. 

Differentiating  both  sides,  we  get 

1  dy      fM 
y  dx      X 

from  which  we  get  1)  at  once.     This  method  rests  on  the  assump- 
tion that  -^  exists,  and  so  is  open  to  the  criticism  of  377,  2. 
dx 

EXAMPLES 

388.    1.  y  =  a  +  b</^^=f(x).         b>0. 

For  a;  >  0,  we  can  apply  387,  getting,  since  here  At  =  f , 

dx      S-^^  "^ 


FUNDAMENTAL   FORMULA   OF   DIFFERENTIATION       241 

2.  For  x<0,  the  formula  of  387  is  inapplicable,  since  it  resta 
on  the  essential  hypothesis  that  a:  >  0.  We  can,  however,  adopt  a 
method  applicable  to  any  x=^0. 

Set  x^  =  u. 

Then  y  =  a  +  hu^. 

For  all  X  in  9?  which  are  =/=0,  u  is  >0. 
Applying  387,  we  have 


du      3 
On  the  other  hand,  by  378, 


hu  ^ 


dy  _dy    dM 

dx      du    dx 


since 

is  finite. 
Hence 


du 
dx 


=  2; 


dy  ^2    b 

dx      3  -^x 


3.  When  rr  =  0,  even  this  method  fails,  as  u  must  be  >  0,  in  order 
to  apply  387.  In  order  to  calculate  the  differential  coefficient  at 
this  point,  we  must  start  from  its  definition. 

We  have,  setting  h  =  Aa;, 

^y    /(A)-/(0)     ,VP 


Aa; 
Here,  when  Aa;  =  0, 


=  h 


R  lim  — ^  =  +  30,  L  lim  -^  =  —  oo. 

Aa;  Aa; 


The  graph  of  f(x)  has  thus  a  vertical  cusp  at  the  origin,  as  in 

Fig.  1.:  , 


242  DIFFERENTIATION 

4.    In  order  to  get 

72/(0),     i/CO), 

some  readers  may  be  tempted  to  take  the  right  and  left  hand 
limits  of  the  expression  1)  for  x=0.  In  the  present  case  we  would 
get  the  correct  result.  In  general,  if  the  expression  for /'(a;) 
assumes  an  indeterminate  form  for  a  particular  value  of  x,  say 
x  =  a,  the  reader  must  avoid  the  temptation  to  conclude  that 

/'(a)=lim/'(a:). 

oo=a 

This  is  only  true  when  /'  (2;)  is  continuous  at  a. 
Ex.1.  f(x)  =  xs\n-,  x^O; 

X 

=  0,  x  =  0. 

Here/'(0)  does  not  exist  by  367,  while,  for  x^^O, 


/'(x)=sinl--cosi. 

XXX 

Thus 
also  does  not  exist. 

lim/'(ic) 

x=0 

Ex.2. 

/(cc)  =  a;2  sin -,  aj^O; 

X 

Here 

=  0,            «  =  0. 

while,  for  a;:^0, 

/'(0)  =  0,  by  368, 

f'(x)  =  2  a;  sin  i  -  cos  1. 

X               X 

Thus 
does  not  exist. 

lim/'(x) 

x=0 

r 


5.  Let  I 

f{x)  =  x^. 

We  find  readily  that 

i2/(0)  =  i/(0)=+oo.  "     J 

The  graph  is  given  in  Fig.  2.  yiq.  2. 


FUNDAMENTAL   FORMULA   OF   DIFFERJ:NTIATI0N       243 
389.    1.  Let        log  x  =  l^^        log  log  x  =  l^x, 

log  log  log  x  =  IgX,  etc. 

Since  log  u  is  defined  only  for  w  >  0,  we  shall  suppose  that  x 
is  taken  sufficiently  large  so  that  l^x  has  a  meaning. 
We  prove  now 

DJr„x  =  - .         m>l.  (1 

J,     til  J  -f  J  V 

y=l^x  =  log  log  x. 
u  =  log  X. 
y  =  log  U. 

dx      du    dx      u   X      X  log  x 
y=l^x  =  log  .  l^x. 


For,  first,  let 

Set 

Then 

Hence 

Next,  let 
Set 
Then 
Hence 

By  2), 

Hence 


u  =  l^x. 

y  =  log  V,. 

dy     dy    du 
dx      du    dx 

du      T\  1        11 

dx  X  log  X 

dx      u    X  log  X      xl^xl^x 
By  induction,  we  now  establish  1)  readily. 

2.  In  a  similar  manner  we  establish 


244  DIFFERENTIATION 

From  1),  3)  we  have  two  formulae  to  be  used  later : 

i>A^-^  =  - -—^ J-;         m>l.  (4 

and 

J>A-^  = 1    ^-^     1 -■     ^*l-     (5 

^  '^^x  —  a  x—a^x  —  a 

In  4),  5)  we  suppose  a;>a,  such  that  the  quantities  entering 
them  are  defined. 


Differentials  and  Infinitesimals 
390.   1.  Since 


/(:.)  =  lim^, 


we  have  for  each  e>0,  a  S>0,  such  that  in  V^^Qc)^ 


^-/'(^) 


or 

I  ^y  —f  (a^)^a:  I  <  e  I  Aa;  |, 
or 

Ay=/'(2;)Aa:  +  e'Aa;,  (1 

where 

I  e'  I  <  e. 

We  call 

fXx)^x 

the  differential  off(x)^  and  denote  it  by 

dy   or   dfQc). 
The  relation  1)  shows  that  A«/  is  made  up  of  two  parts,  viz, 

dy   and   e'Aa;. 
The  ratio  of  these  two  parts  is 

,  — ^.  f(x^^Q. 


Q 


DIFFERENTIALS   AND   INFINITESIMALS  245 

As /'(a;)  is  fixed,  for  fixed  x,  and  e'  can  be  made  numerically 
as  small  as  we  please,  by  taking  h  sufficiently  small,  we  see  that 
the  part  e'Aa;  is  very  small,  compared  with  dy  for  all  points  x-\-/\x 
in  Fg*.  Thus,  in  the  immediate  vicinity  of  x,  the  principal  part 
of  Ay  is  di/. 

Differentials  owe  their  importance  to  this  fact. 

2.  To  make  the  notation  homogeneous,  it  is  customary  to  replace 
Ax  by  another  symbol,  dx,  in  the  expression  for  dy.     We  have  then 

dy  =/'  (^x^dx, 

391.  The  notion  of  a  differential  may  be  illustrated  as  follows : 
Let  the  graph  of  /(a;)    be    that  in   the 

figure. 

Let  PB,  be  the  tangent  at  P;  and 

PS=Ax,    QS=Ay,   RS=dy. 

QB  =  e'Ax. 

The  reader  will  see,  if  dy^Q^  that  as 
Q  approaches  P,  QR  becomes  smaller  and  smaller  as  compared 
with  RS=dy.  This  is  illustrated  by  comparing  this  ratio  at  Q 
and  at  Q' . 

We  see  dy  =  RS  approximates  more  and  more  closely  to  Ay  as 
Q  approaches  P. 

392.  A  variable  whose  limit  is  0  is  called  an  infinitesimal. 
When  employing  differentials,  we  suppose  that  the  increment 

given  to  the  independent  variable  Ax  =  dx  can  be  taken  as  small, 
numerically,  as  we  choose.  It  is  thus  an  infinitesimal.  Then 
both  Ay  and  dy  are  also  infinitesimals. 

In  the  limits  considered  in  301-304,  310-312,  the  numerators 
and  denominators  furnish  examples  of  infinitesimals. 

Also  the  lengths  of  the  intervals  considered  in  127,  2,  are  infini- 
tesimals. 

Many  other  examples  of  infinitesimals  are  to  be  found  in  the 
preceding  pages,  and  many  more  will  occur  in  the  following. 


246 


DIFFERENTIATION 


The  Law  of  the  Mean 

393.  One  of  the  pillars  which  support  the  modern  rigorous 
development  of  the  calculus  is  the  Law  of  the  Mean.     It  rests  on 

Holies  Theorem.  Let  f(x)  he  continuous  in  31  =  (*?  ^)»  and 
fi_a)=f(J>).  Let  f  (x)  he  finite  or  infinite  within  51.  Then  there 
exists  a  point  c  within  31,  for  which 

/(c)  =0.         a<e<h. 

Since  f(x)  is  continuous  in  3t,  it  is  limited,  by  350.  Its  ex- 
tremes are  therefore  finite.  If  /  is  not  constant,  one  of  these 
extremes  is  different  from  the  end  values. 

To  fix  the  ideas,  let  Max/  =  yn  be  different  from  the  end  values. 

By  354,  there  is  a  point  c  within  31  such  that 

while  for  all  points  c  +  A  of  31,  A  >  0, 

/(e+A)-/(c)<0,    /(c-A)-/(c)<0. 
Hence 

Kc  +  h^-fjc-)^ 

h       -  ' 

/(^-A)-/(O^Q^ 
—  h 

/(c)^0; 

Those  together  require  that 

/'(.)=  0. 
In  case  f(x)  is  a  constant  in  31,  the  theorem  is  obviously  true 

394.  1.  The  geometric  interpre- 
tation of  Rolle's  theorem  is  the 
following : 

Let  the  graph  of  f(x)  be  a  con- 
tinuous curve  having  everywhere 
a   tangent,  except   possibly  at   the 


(1 
(2 


According  to  1), 
according  to  2), 


1 


THE   LAW   OF   THE   MEAN  247 

end  points  J.,  -B,  which  are  at  the  same  height  above  or  below  the 
a;-axis.     Then  at  some  point  O  the  tangent  is  parallel  to  the  rc-axis. 
Since /'(a;)  may  be  infinite,  the  graph  may  have  points  of  inflec- 
tion with  vertical  tangents,  as  at  P. 

2.  Let  A^  B  be  two  points  at  the  same  height  above  the  a;-axis. 
The  reader  will  feel  the  truth  of  Rolle's  theorem  for  simple  cases 
if  he  tries  to  draw  a  continuous  curve  V  through  A,  B,  whose 
tangent  is  not  parallel  to  the  cc-axis.  F  should,  of  course,  have  no 
vertical  cusp  or  angular  point.  We  say  for  simple  cases,  because 
we  cannot  draw  a  curve  with  an  infinite  number  of  oscillations  or 
a  curve  which  does  not  have  a  tangent  at  A  or  B.  Yet  neither  of 
these  cases  need  to  be  excluded  in  Rolle's  theorem. 

395.  lif'(x)  does  not  exist  for  some  point  within  %  the  theorem 
394  is  not  necessarily  true,  as  Fig.  1  shows.     (See  366.) 


1 

k 

a 

b 

Fig.  1.  Fig.  2. 

lif'(x)  is  not  continuous  in  21,  the  theorem  does  not  need  to  be 
true,  as  Fig.  2  shows. 

396.  1.  Criticism.  Many  demonstrations  are  rendered  invalid 
because  they  rest  on  the  assumption : 

1°.  In  passing  from  a  to  J,  the  function  must  first  increase  and 
then  decrease,  or  first  decrease  and  then  increase ; 

or  on  the  assumption  : 

2°.  There  must  be  at  least  one  point  between  a,  h  where  the 
function  ceases  to  increase  and  begins  to  decrease,  or  conversely. 

Either  of  these  assumptions  is  true  if  we  use  functions  having 
only  a  finite  number  of  oscillations  in  31. 

In  case  the  function  has  an  infinite  number  of  oscillations  in  51, 
neither  of  the  above  assumptions  need  be  true. 


248  DIFFERENTIATION 

The  function  of  Ex.  2,  378,  where  51  =  (0,  1),  illustrates  the 
untruth  of  1°. 

We  shall  later  exhibit  functions  which  oscillate  infinitely  often 
in  any  little  interval  of  31  and  yet  have  a  derivative  in  51-  Such 
functions  show  that  2°  is  not  always  correct. 

2.  The  demonstration  given  in  393  is  extremely  simple.  It 
rests,  however,  on  the  property  that  a  continuous  function  takes 
on  its  extreme  values  in  an  interval  (a,  6).  In  an  elementary 
treatise  this  fact  might  be  admitted  without  proof,  since  it  seems 
so  obvious. 

397.    1.  Laiv  of  the  Mean.     Let  f(x)  he  continuous  in  51  =  («,  ^), 
and  letf'(x)  he  finite  or  infinite.,  witlmi  21. 
Then,  for  some  point  a<cc<.h., 

/(^) -/(«)=  (^-«)/(0-  (1 

Consider  the  auxiliary  function 

9ix)  =f(h)  -fix-)  -^^^l  -{^^>  (h  -  :.). 

Evidently 

^(a)=(/(5)=0. 

Also  at  those  points,  at  which /'(a;)  is  finite, 

/(.)=-/(.)  +  ^W9^;  (2 

while  at  the  other  points  of  51,  g' (x)  is  infinite.  Thus  gQx")  is 
continuous  in  2t,  and  g'Qjc)  is  finite  or  infinite  within  %.  Hence, 
for  some  point  a<c<.h, 

g'ic)=%      by  393.  (3 

Setting  x=  c  m  2),  and  using  3),  we  get 


/(,)  =  /(^)-/W  (4 

^  ^  b  —  a 


which  is  1). 


THE   LAW   OF   THE   MEAN 

2.   The  relation  1)  is  commonly  written  as  follows 
Set  h—  a  =  h; 

then,  since  c  lies  within  (a,  J), 


249 


Thus  1)  gives 


f(a  +  li)  =f(ia')  +  /^'  (a  +  ^A). 


3.  The  reader  should  observe  that  although  /'(x)  may  be 
infinite  within  21,  the  point  c  in  1)  is  such  that /'(c)  is  finite. 

398.  The  following  is  the  geometric  interpretation  of  the  Law 
of  the  Mean.  Let  ACB  be  the  graph  oif(x)  in  51-  Let  the  chord 
AB  make  the  angle  6  with  the  rr-axis.     Then 

ta„^==«*)-«2). 
0  —  a 

Also  by  397,  4), 

/(c)=tan(9. 

That  is :  at  some  point  c  within  the 
interval  (a,  h)  the  tangent  is  parallel  to  the  chord  AB. 

399.  When  either  of  the  conditions  that  enter  the  Law  of  the 
Mean  are  violated,  the  point  c  may  not  exist.  This  is  illustrated 
by  the  following. 


Fig.  1. 


Fig.  2. 


Fig.  3. 


1.  f(pc)  is  not  continuous  in  (a,  H).     Fig.  1. 

2.  The  differential  coefficient  does  not  exist  at  some  point  within 
(a,  h).     Figs.  2,  3. 


250  DIFFERENTIATION 

400.  We  give  now  some  elementary  applications  of  the  Law  of 
the  Mean. 

1.  Let  f{pc)  he  continuous  in  5t  =  («,  5);  and  let  its  derivative 
f'(x)  =  A,,  a  constant,  within  %.      Then 

f(x}  =  \x  +  ix,     in  51.  (1 

Let  a;  >  a  be  a  point  of  21.  The  function  /(a;)  satisfies  the  con- 
ditions of  the  Law  of  the  Mean  in  :53  =  (a,  x}. 

Hence,  by  397, 

/(a;)  =/(«)  + (a;- a)/ (c).         a<c<x.        (2 

Since  by  hypothesis, 

we  get  1)  from  2),  on  setting 

/t.=/(a)-a/(.). 

Since,  by  hypothesis,  /(a;)  is  continuous,  the  formula  1)  is  also 
true  for  x=  a. 

2.  As  a  corollary  of  1  we  have : 

Let  fix)  he  continuous  in  %  =  (a,  6)  and  let  its  derivative  he  0 
within  21.      Then  f(x)  is  a  constant  in  21. 

3.  Let  f(x),  g(x)  he  continuous  in  the  interval  21,  and  let  f  (x) 
=  g'  (a;)  within  %.      Then,  O  heing  some  constant, 

/(a;)=^(a;)+(7,     in  2[.  (3 

For 

satisfies  the  conditions  of  2.     Hence 

k(x)=0,     in  21. 

401.  Let  f(x)  he  continuous  in  21  =  (a,  J),  while  f  (x)  is  finite  or 
infinite  within  21.  Letf'Qc^,  when  not  0,  have  one  sign  a.  Then  f 
is  monotone  increasing  in  21,  if  o-  is  positive;  monotone  decreasing,  if 
(T  is  negative. 

Let  a<x'  <x"  < h. 


THE   LAW   OF   THE   MEAN  251 

By  the  Law  of  the  Mean, 

f(x")=f{x'')  +  (x"-x'-)f'ic').         x'<c<x". 
As  x"  —x'  >0,  and  /'(c)  has  the  sign  cr,  when  not  0, 

fix"}  ^/(a;'),  if  o-  is  positive  ; 

/(a;")</(a;'),  if  cr  is  negative. 

402.  Criticism.  Some  writers  state  that  f(x)  is  increasing  when 
/'  (x)  is  positive,  and  conversely  when  f(x)  is  increasing  /'  (x)  is 
positive. 

The  second  statement  is  not  true,  as  the  figure  shows.  P  is  a 
point  of  inflection,  with  a  tangent  parallel  to  the  a;-axis. 

The  error  in  the  reasoning  is  instructive.  By  definition,  if /(a;) 
is  increasing, 

^x 
is  positive.     It  is  now  inferred  that  therefore 

Urn  1^  =/'(.) 
is  positive.      All  one  can  strictly  infer  is  that 

The  function 

y  =  x?,         31  =  (-1,1) 

illustrates  this,  at  the  point  a;  =  0.     See  Fig. 

403.  Let  f(x)  he  continuous  in  21  =  (a,  5),  and  f  {x)  finite  or 
infinite  within  %.  f  (x)  shall  not  vanish  for  all  the  points  of  any 
subinterval  Sd  =  («,  ^)  of  21.  When  not  0,  letf'(x')  have  always  one 
sign  (T  in  21.  Thenf(x)  is  an  increasing  or  a  decreasing  function 
in  21,  according  as  a  is  positive  or  negative. 

To  fix  the  ideas,  let  a-  be  positive. 
Let  a<x'<x"<b. 

By  the  Law  of  the  Mean, 

/(x")  =/(a;')  +  Qx" -  x')f{c').       x'<c<x". 


T 


252  DIFFERENTIATION 

Kq  x"  —  x'>  0,  and  /'  (c?)  ^  0,  we  have 

f{x")>f{x'y, 

i.e.^f(jc)  is  monotone  increasing  in  31-     To  show  it  is  constantly 
increasing  in  21,  suppose 

Then /(a;)  must  =f(a)  for  all  points  in  ^  =  («,  /3),  since  it  is  a 
monotone  increasing  function. 

Since /(a;)  is  a  constant  in  SQ,f'(x)  =  0  in  ^,  which  contradicts 
the  hypothesis. 

404.  Let  f'^x^  be  eofitinuous  in  the  intei'val  2(.  Then  the  differ- 
ence quotient  — ^  converges  uniformly  tof'Qx}  in  %. 

Ax 

For,  by  the  Law  of  the  Mean, 

f{x  +  h~)-f(x)=hf(x+eh').         0<6'<1. 
Hence  Ay 


Ax 


^f'(x  +  eh^.  (1 


But /'(a;)  being  continuous  in  2t,  is  uniformly  continuous  by 
352.     Hence 

lim/'  (x  -{-dh')=  f  (x) .      uniformly . 

ft=0 

'    -^    ^'  lim -r^=/'(x).       uniformly  in  %. 


Derivatives  of  Higher  Order 

405.    The  first  derivative  of/' (a;)  is  called  the  second  derivative 
of  f(x^,  and  is  denoted  by 


Evidently, 


/'(a;),  J)Jfix},  1^. 
dx^ 


ft=o  A  ^^=0     ^a; 

this  limit  being  finite  or  infinite. 


DERIVATIVES   OF   HIGHER   ORDER  253 

In  this  way  we  may  continue  to   form  third,  fourth,  ...  and 
derivatives  of  any  order. 

Derivatives  of  order  n  are  denoted  by 

406.  We  add  the  following  formulae,  which  will  be  used  later. 

They  are  easily  verified  : 

i)X=/i./x-l.  .../A-w  +  l.a;'*-".          x>Q.  (1 

I)l(l  +  xy  =  ^i'ix-l....fi-n  +  l-(l  +  xy-''.  l+x>^.        (2 

i>>^  =  e^.  (3 

D^  sin  X  =  sin  [  —  +  x\.  (4 

2)!^cosa;=  cosf  ^  +  a;].  (5 

407.  Let  y  =  mw,  where  w,  v  have  derivatives  of  any  desired 
order.     The  following  relation  is  known  as  Leibnitz' s  formula. 

where  fn\     n  •  n  —  \  •  •••n~m-\-\ 


mj  1.2  —  m 

We  prove  it  by  complete  induction;  i.e.  we  assume  it  true  for 
n  and  prove  it  is  true  for  n  +  1.     For  n  =  1,  2  it  is  obviously  true. 
Differentiating  1),  we  get 


254  DIFFERENTIATION 

Now,  by  96, 

\mj     \m  —  lj     \    m   J 
This  in  2),  gives 

which  is  1),  when  we  replace  in  it  nhy  n-\- 1. 

408.    1.  Let  us  apply  Leibnitz's  formula  to  find  the  derivatives 

of 

y  =f(x)  =  tan  X. 

We  have 

y  =  sec'^a;, 

1/"  =  2  secure  tan  x  =  2  t/y'. 
Now 

This  gives 

y"=2(y'3/+y2). 

3/-  =  2(t/*^2/  +  4  y'"y  +  3  y"2),  etc. 

2.  Another  way  is  the  following,  which  will  lead  us  to  a  formula 
that  we  shall  need  later. 
We  have 

1/  cos  X  =  sin  X ; 

or  setting, 

u  =  sin  X,  z  =  cos  a;, 

u  =  yz. 
Now  by  Leibnitz's  formula, 


TAYLOR'S   DEVELOPMENT   IN   FINITE    FORM  255 

Also,  by  406,  4),  5), 


sin  X. 


Hence  1)  gives 

sin  (~  +  ^)={  ^'"^  -  (2)^^"-^'  +  (^)^<"-^'  -  • 

This  gives  the  recursion  formula, 

sin  f  -—  +  a;  ]  ^n 

+  tana;  I  (!^')y«-i'-('^')y-«>  +  ...  I . 
Setting  a;  =  0  in  2),  we  get 

/»)(0)-(^)/(-2>(0)  +  (^^)/-«(0)--.=  sin^.  (3 

Taylor  s  Developinent  in  Finite  Form 

409.  1.  Using  derivatives  of  higher  order,  we  can  generalize 
the  Law  of  the  Mean  as  follows : 

In  the  interval  %  =  (a,  5),  let  f{x)  and  its  first  n  —  1  derivatives 
be  continuous.  Let  f^"\x)  he  finite  or  infinite  within  %.  Then  for 
any  x  in  51, 

/(2;)=/(a)  +  ^^^/(a)  +  ... 

n~ll  nl 

where 

w  a<e<x. 


a56  DIFFERENTIATION 

As  in  397,  we  introduce  an  auxiliary  function 

giu-)  =/(:.)  -/(^)  -ix-  u~)f'(u}  -  (^^^f"(u)  - ... 

w  —  1 !  nl 

where  A  is  independent  of  u.  This  function  is  obviously  a  con- 
tinuous function  of  u  in  21  for  any  x  in  St.  Differentiating  with 
respect  to  u,  we  get,  observing  terms  cancel  in  pairs, 

9'(u:,  =  (^^ZJDTI  s  -/")(^)  +  AI  (3 

71—11 

for  any  u  tvithin  21. 

Thus  the  derivative  of  g{u')  is  finite  within  21. 

To  apply  Rolle's  theorem  to  g(u)-,  with  reference  to  the  interval 
^  =  (a,  a;),  it  is  only  necessary  to  determine  A  in  2)  so  that 

But  obviously 

g(x')=0. 

We  therefore  suppose  A  so  chosen  that 
0  =f(x-)  -fCa-)  -(x-  a)f\a^  -  •.• 

-  -fcl^)^/(-i)(«)  -  (^z:^  A.  (4 

w— 1 !  w 1 

Then  by  Rolle's  theorem,  there  is  a  point  a<c<x^  such  that 

This  in  3)  gives 

^^^^f/">(c)-^l  =  0.  (5 

M—  1 1 

As  c^^a;,  the  first  factor  in  5)  is  not  0.     Hence  the  parenthesis 
is  0,  which  gives 

r^\c)=A. 

Putting  this  value  of  A  in  4),  we  have  1). 


TAYLOR'S   DEVELOPMENT   IN   FINITE   FORM  257 

2.  The  formula  1)  is  called  Taylor  s  development  of  f(cc)  in  finite 
form. 

It  may  also  be  written  as  follows : 

Set 

x  =  a+h,     c  =  a-\-6h.  0<0<1. 

Then  1)  becomes 

/(a  +  A)=/(a)  +  l/'(a)  +  |^/"(a)  +  ... 

+  ~^/'"-"(<^)  +  ^fKa  +  6h).  (6 

n  —  11  ni 

a-\-h,  in  21. 

410.  1.  Letf(x)  and  its  first  n  —  1  derivatives  be  continuous  in 
the  interval  SQ  =  (a  —  H^  a  +  ZT),  while  f^'^^x)  is  finite  in  ^.  Then 
for  any  x  in  iS, 

+  (^-^)"">-i)(a)  +  ^^=^>«)(c),  (1 

w  —  1 !  n\ 

=/(a  +  A)  =/(a)  +  A/' (a)  + 1!/'' («)  +  ... 


where 


+  -^/"-'^(a)  +  A/<")(a  +  dh-),  (2 

n  —  1\  n  : 

x  =  a  +  h,     c  =  a  +  6h,     0<^<1,     \h\<H. 


The  truth  of  this  theorem  for  the  left  hand  half  of  ^  follows 
from  the  fact  that  the  reasoning  of  409  does  not  depend  upon  a 
being  <  6  ;  it  holds  when  a  >  5,  if  we  change  x  and  c  accordingly. 

2.  When  a  =  0,  1)  gives 

fix)  =/(0)  +  f-/'(0)  +  f'/'co)  +  -.  +  "^^r-Kdx).      (3 

1  I  2 !  nl 

Q<d<l. 
This  is  known  as  Maclaurin's  development. 


258  DIFFERENTIATION 

411.    1.   Let /(a;)  =  sin  re. 
From  410,  3)  we  get 

osin  Qx  ^^ 

^\XV  X  =  X  —  x'' —-- —  (1 

2  ! 

=  a;  -  —  +  -^sin  Bx,         0  <  ^  <  1. 
d  !       4  ! 

ryO  /i^O  /mD 

etc. 

The  ^'s  in  these  formulse  are  not  necessarily  the  same. 
Let 


^<x< 

2* 

These  formulae  show  then  that 

sin  x<x 

>x- 

0? 

3! 

<x- 

3! 

+  6!' 

etc. 
From  1)  we  have  again 


See  801. 

412.    Let 

As  before,  we  get 


T     sin  a;      -,       T           sin  ^a;      ^ 
lim =  1  —  lim  X —  =  1. 

a;=o      a;  »=o  2 

f(x)  =  cos  a;, 
cos  a;  <  1 


From 


^        2!' 

0<.<| 

<  1  _  ^  +  ^* 

2!^4!' 

etc. 

-,       a;2   ,    a:^ 

sin 

^a;, 

PARTIAL   DIFFERENTIATION 


259 


we  have 


lim  — 

x=0 

X  _  ± 

~  ~2~ 

-  — -  lim  X  sin  dx 

3 !  x=o 

See  304. 

=i- 

413.   1.  Let 

From  410,  3)  we  get 

Ic 
Hence,  as  in  310, 

2.  Let 


/(x)=log(H-a;). 


From  410,  3)  we  have 


f(x)  =  a^         a>  0. 


a"^  =  1  +  —  log  «  +  ^  log^  a  '  a^^. 


Hence,  as  in  311, 
3.  Let 


a^  —  1 
lim =  log  a. 

x=0  X 


f(ix~)  =  (\^xY. 


We  get  from  410,  3) 
y^xy=l-\-fJLX-\- 
Hence,  as  in  312, 


(i  +  xy  =  i  +  fix  +  a?' ^'^   ^(1  +  exy-\ 


0<d<l. 


0<^<1. 


|a:|<l,  0<6'<L 


lim  ^^ — ■ — =  fi. 

x=0  X 


FUNCTIONS  OF  SEVERAL  VARIABLES 

Partial  Differentiation 

414.  The  definition  of  a  partial  differential  coefficient  and  a 
partial  derivative  of  a  function  of  several  variables  is  analogous 
to  the  corresponding  definitions  for  functions  of  a  single  variable. 


260  DIFFERENTIATION 

Let/(a;j-"2:^)  be  defined  over  a  domain  i),  for  which  x—  (x^-'X,,^ 
is  a  proper  limiting  point. 

Let  x^  =  (x^'--x^.x^  x^+  A,  x^+i-'-x^}  be  any  point  of  D,  different 
from  X.     If 

A=0  fl 

is  finite  or  infinite,  it  is  called  the  (first)  partial  differential  coeffi- 
cient off  with  respect  to  x^  at  the  point  x.     The  aggregate  of  these 
rfs,  defines  a  new  function  over  a  certain  domain  A^i),  which  is 
called  the  (first)  partial  derivative  off  tvith  respect  to  x^. 
It  is  denoted  variously  by 

-^j:,J\.^l'"^m)i     1 ^— »  /j-,(^l*"^m)'  (^ 

ox. 

When  h  in  1)  is  restricted  to  positive  values,  tj  is  called  a  right 
hand  partial  differential  coefficient,  and  their  aggregate  gives  rise 
to  the  right  hand  partial  derivative  with  respect  to  x,. 

The  meaning  of  the  terms  left  hand  partial  differential  coefficient 
and  derivative  with  respect  to  x,  is  obvious. 

They  are  denoted  by  putting  the  letters  R  and  L  before  the 
symbols  2). 

The  function  f{x^  •  •  •  a:„)  has  therefore  in  general  m  (first)  par- 
tial derivatives, 

-^,    ^,    ...    -^. 
dx^      dx^  dx„^ 

The  process  of  obtaining  these  partial  derivatives  is  called 
partial  differentiation. 

415.    Example. 


f(xy)  =  Vx2  +  2/2. 
If  the  point  x,  y  is  not  at  the  origin, 

a;2  +  2/2  >  0. 

We  can  therefore  apply  378  and  387,  getting 

df  X  df  _ 


dx      Va;2  +  2/2     dy      Va;2  +  j/2 

Thus  the  partial  derivatives  with  respect  to  x  and  y  exist  at  all  points  different 
from  the  origin. 


PARTIAL   DIFFERENTIATION 


261 


When  the  point  x,  y  is  at  the  origin,  we  cannot  apply  this  method.     (Compare 
388.) 

We  therefore  proceed  directly.     We  have 


This  shows  that 


A%        Ax 

i?/',(0,  0)=r+l,     Z/,(0,  0)=-l. 


Thus  the  partial  differential  coefficient  with  respect  to  x  does  not  exist  at  the 
origin.     Similarly, 

Bfy{Q,  0)  =  +  1,    Z/^(0,  0)  =  -  1  ; 

and  the  partial  derivative  with  respect  to  y  does  not  exist  at  the  origin. 

416.  In  the  case  of  two  independent  variables,  the  (first) 
partial  differential  coefficients  admit  a  simple  geometric  inter- 
pretation. 

Let  the  graph  of 

z=f(x,  y') 

be  a  surface  S.     The  plane 

y  =  constant 

intersects  ^  in  a  curve  C. 

Let  PT  be  the  tangent  to  C  at  P  =  (a:,  y,  2),  making  the  angle  6 
with  the  a;-axis.     Then 


bx 


=  tan  6. 


Compare  365. 

The  partial  differential  coefficient 

dy 

has  a  similar  meaning  with  respect  to  the  y-a.xis. 


417.  Let  f^  (a:^  •"^m)  be  finite  for  a  domain  A.  We  may  now 
reason  on  /j.  as  we  did  on  /.  Let  a^  be  a  proper  limiting  point  of 
A,  and  x'  =  {x^  •  •  •  Xj_i,  x^  +  h,  x^+i  •■■x,^  any  point  of  A  different 
from  X. 


262  DIFFERENTIATION 

ft=o  h 

is  finite  or  infinite,  77  is  called  the  second  partial  differential  coeffi- 
cient of  /  with  respect  to  2;^,  x^  at  the  point  a;,  and  is  denoted  by 

7^2           -C/'                         ~\           •/  V     1    *  *  *      771 J         -Pf        /"  \ 

■^  XjaTj/C^l  •  •  ■  ^my »    «      a i    •^ ^i^j  V^l  *  * '  •^'n-' ' 

The  aggregate  of  these  t/'s  will  define  a  new  function  over  a  cer- 
tain domain  Aj  ^  A,  which  is  called  the  second  partial  derivative^ 
first  with  respect  to  x^,  then  with  respect  to  Xj. 

Proceeding  in  this  way,  we  may  form  third,  fourth,  •••  partial 
differential  coefficients  and  derivatives. 

Change  in  the   Order  of  Differentiating 

418.  1.  In  almost  all  cases  which  occur  in  practice,  the  partial 
differential  coefficient  has  the  same  value,  however  the  order  of 
differentiation  is  chosen.     For  example  : 

f"     =  /'"     =  f"     =  /'"     =  f"     =  /'" 

That  this  is  not  always  true  is  shown  by  the  following  example : 

2.  f(^iy)—'^y~^ — ^'    for  points  different  from  the  origin. 

X  -\-  y 

=  0,    for  the  origin. 
Then  if  x,  y  is  not  the  origin, 

bx      ^\x^  +  y'^      (x^  +  y'^yy  ^ 


^  =  x[  ^^ "~  ^^  -      "^^V      I  (2 


dy         [x^  +  y^      (2-2  +  2/2)2 
At  the  origin, 


^=0,   ^=0.  (3 

dx        '    By  ^ 


CHANGE   IN   THE   ORDER  OF   DIFFERENTIATING         263 
From  1),  2)  we  have,  in  particular, 

Consider  now  the  second  partial  derivatives. 

From  1),  2)  we  have,  for  all  points  different  from  the  origin, 

^  _  :i;2  _  ^2  r  8a:y     1  _    dj 


dxdy      x^  +  y^  \         {pfi  +  y'^)^  j       dydx 
At  the  origin,  we  have  from  3),  4),  5), 

^=-^;    .•./-(0,0)=-l.  (6 

|S  =  ^;    .../-(0,0)=  +  l.  (7 


Hence,  at  the  origin, 


52/  ^dj 


dxdy     dydx 


3.  In  connection  with  this  example,  we  may  warn  the  inexperi- 
enced reader  to  avoid  certain  errors  he  is  likely  to  fall  into. 
To  get  the  equations  3),  i.e. 

/X0,0)=0,  /X0,0)  =  0, 

it  is  not  permissible  to  set  a:  =  0,  y  =  0  in  the  relations  1),  2).     In 
fact,  these  formulae  were  obtained  under  the  express  stipulation 
that  this  point  x  =  y  =  Q  be  ruled  out. 
To  get  the  equation  6),  i.e. 

/^(0,0)  =  -l,  (6 

it  is  not  permissible  to  differentiate  4),  i.e. 

fXO,y)  =  -^,  (4 

with  respect  to  y,  thus  getting 

and  in  this  set  y=0,  getting  the  required  value  of /^(O,  0). 


264  DIFFERENTIATION 

In  fact,  the  relation  4)  was  obtained  under  the  express  condition 
that  y  =5^  0. 

Similar  remarks  apply  to/^j^(0,  0). 

Junior  students  are  so  accustomed  to  differentiate  with  their 
eyes  shut  that  they  often  overlook  the  fact  that  formulse  and 
theorems  are  usually  not  universally  true,  but  are  subject  to 
more  or  less  stringent  conditions.  Compare  also  the  example 
of  388,  4. 

419.   It  is  easy  to  see  a  priori  why  /^^(a,  5)  mai/  be  different 
from  fly'^i^ah). 
By  definition, 

/i.(«.  2/) = ii™  — S ' 

A=o  n 

4=0  K 

..     If,.     fCa  +  h,b  +  k')~f(a,b  +  k) 
=  lim  -  \  lim  ^^ ^ — ^^^ 

fc=0   rC  [    71=0  n 

h^  h  } 

Let  us  set 

^.,    ,,^     /(g  +  Kb  +  k)  -fja.  b  +  h) -fja  +  A,  5)  +/(a,  5) 

Then  1)  gives 

/i;(a,  6)  =  lim  lim  FQi,  k~). 

In  a  similar  manner  we  find  that 

fy^cQa,  b}  =  lim  lim  F{h,  Tc). 

7i=0      k=a 

These  formulae  show  that  /^y(a,  6),  fyxia,  5)  are  double  iterated 
limits,  taken  in  different  order.  It  is  therefore  not  astonishing 
that  a  change  in  the  order  of  passing  to  the  limit  may  produce  a 
change  in  the  result.     Cf.  322,  323. 


1 


CHANGE   m   THE   ORDER  OF   DIFFERENTIATING         265 

420.  1.  We  consider  now  certain  cases  when  it  is  possible 
to  change  the  order  of  differentiation  in  a  partial  differential 
coefficient. 

Let  f(xy')  he  defined  in  DQa,  5).  Let  L*  be  the  deleted  domain 
of  JD.      We  suppose  : 


a)  that  fl  exists  in  7), 

)8)  that  f'Jy  exists  in  2>*, 

7)  that  lim/^^  =  A,.         finite  or  infinite. 

x=a,y=b 

fg{a,b)=\. 

a 

Then 

If,  moreover, 

S)  fl  exists  for  all  points  of  D  on  the  line  y  =  h  ;  then 

f;,'Xa,b)  =  \.  (2 

We  suppose  first,  that  all  four  conditions  a-8  are  satisfied,  and 

show  that  then  „,,.     .         ^.,       ■, 

f[',Ca,b:,=f';Aa,b}.  (2' 

Let 

^     /(a  +  h,b  +  k)  -f(a.  h  +  k)  -fia  +  A,  b}  +fiab) 
^= hk ' 

as  in  419.     We  introduce  the  auxiliary  functions 

aCx)=f(x,b+k}-f(x,b%  (3 

Hiy^^fia  +  Ky^-fia,y-).  (4 

^^^^  hkF  =  aCa  +  h)  -  a^a}  (5 

=  H{b  +  k')-H(b^.  (6 

Setting,  as  usual, 

h  =  Aa:,     k  =  A2/, 

we  have  from  S), 

A/ 


^^ -/;(-,  ^) 


=  e(a;); 


where  e  =  e(a;)  is  a  function  of  k  and  x,  such  that 

lim  e  =  0.  (7 


fc=0 


266  DIFFERENTIATION 

Similarly,  by  a), 

'  Hiy^^MMa.y^  +  rtiy'yU  (9 

where  r}  =  »;(y)  is  a  function  of  h  and  ^,  such  that 

lim  77  =  0.  (10 

ft=0 

Then  5),  8)  give 

F=\\fl,{a  +  hh^-fl,(a,b:^  +  eia  +  K)-e(a)l  (11 

Similarly,  6),  9)  give 

F=\\f':,(ia,h  +  k~)-f'Xa,h-)  +  7j(h  +  k-)-7^(h')\.  (12 

On  the  other  hand,  we  can  apply  the  Law  of  the  Mean  to  5),  by 
virtue  of  a),  getting 

kF=Gr'{c~).         a<c  <a  +  h,  OT  a  +  h<c<a.  (13 
Differentiating  3),  we  get,  using  13), 

kF=\f^ic,h  +  k}-fXo,h}l  (14 

By  virtue  of  /3,  we  can  apply  the  Law  of  the  Mean  to  14), 

getting    ^^^,,  ^^^  ^y          h<d<h  +  k,  ov  h  +  k<d<h.  (15 

From  11),  15)  we  have 

From  12),  15)  we  have 

^ur^    ..^f'M^h  +  k^-fXah-)      v(b  +  k}      v(h-)  ,.„ 

JxyV'^')  aj—  ^  k  k    ' 

We  can  now  apply  324  to  16). 

Now,  by  7),  li:n/iKc,i)=X. 


CHANGE   IN   THE   ORDER   OF   DIFFERENTIATING         267 

^    ^  lime(a  +  A)  =  0,    lime(a)  =  0. 

Hence,  letting  k  first  pass  to  the  limit,  and  then  A, 

»=0         *=0     L  fi  fi  '*      J 

ft=o  h 

=/;K«'  b>  (18 

Similarly,  17)  gives,  letting  first  h  pass  to  the  limit,  and  then  k, 
\=fjyia,h').  (19 

The  equations  18),  19)  prove  2'). 

2.  If  we  wish  to  prove  1),  without  imposing  the  condition  S), 
we  have  only  to  observe  that  17)  has  been  established  without 
reference  to  3).  But,  as  has  just  been  shown,  we  can  conclude 
1)  from  17). 

3.  It  is  well  to  note  that  this  demonstration  does  not  postulate 
the  existence  of  fyj.\  or  the  continuity  of  either  of  the  second 
partial  derivatives ;  or  the  continuity  of  f'y  in  D  or  D*. 

We  observe  also  that  x  and  y  can  obviously  be  interchanged  in 
the  statement  of  the  above  theorem. 

421.  The  case  which  ordinarily  arises  is  embodied  in  the  fol- 
lowing corollary  : 

Let  fl^  f'y^  fly  he  continuous  in  the  domain  of  the  point  a,  h. 
Then  fyxici'-)  i)  exists,  and  is  equal  to  f'Jy(a-,  b}. 

422.  It  is  easy  to  generalize  421  as  follows : 

Let  the  partial  derivatives  of  f(x^  •  •  •  a;^)  of  order  ^n  be  continu- 
ous in  the  domain  of  the  point  x.  Then  we  can  permute  the  indices 
L  in 

without  changing  its  value. 


268  DIFFERENTIATION 

Since  any  permutation  of  the  n  indices 

l'l'  ••'  l' 
12  ?* 

can  be  obtained  from  any  other  permutation 

by  repeated  interchanges  of  successive  indices,  we  have  only  to 
show  that  we  can  interchange  any  two  successive  indices  as  t^,  t^+i 
in  1)  without  changing  its  value. 

Let  us  introduce  the  function  of  x, ,  x, 

'■r        'r+1 

where  we  consider  all  the  variables  on  the  right  as  fixed,  except 
the  two  noted  in  g. 

Then,  by  421,  ^^  ^^ 


dx,  dx,  dx,     dx^ 

'r        '■r+1  'r+1        V 

Hence 

^(r+l)  _  f(r+l) 

J  il'"tr— I'r'r+l         "^  4"V-l'T+l'r' 

Differentiating  now  with  respect  to  x^^  ^•••a;^,  in  the  order 
given,  we  get 

fi.n)  _   fin) 

Totally  Differentiable  Functions 

423.  1.  If  the  function  fQc)  has  a  finite  differential  coefficient 
at  a;  =  a,  we  saw  that 

where  h  is  an  increment  of  x^  and  a  is  a  function  of  A,  such  that 

lim  cc  =  0. 

Under  certain  conditions,  to  be  given  later,  an  analogous  theorem 
holds  for  functions  of  several  variables.  Let  A/  be  the  increment 
that  f(x-^  •••  Xj^  receives  when  we  pass  from  the  point  a=  (a^  —  a,„) 
to  the  point  a  +  A  =  (a^  +  /i^  •••  a„^  +  A„j). 


TOTALLY   DIFFERENTIABLE  FUNCTIONS  269 

Here  any  of  the  A's  may  =  0.     Let 

A/"  =/4(«)^i  +  -  +fL(j^^^m  +  «i^i  +  •"  +  «m^m. 
where  the  a^  are  functions  of  Aj  •••  A^,  such  that 
lim  ttj  =  0,   •••  lim  a^  =  0. 

ft=0  ft=0 

The  function  /  is,  in  this  case,  said  to  be  a  totally  differentiahle 
function  at  a. 

We  call  df=f^^(a)h,  +  -  +/L(«)A.  (1 

the  total  differential  of  f  at  a. 

Thus,  when  /  is  totally  differentiahle  at  a,  A/  consists  of  two 

parts,  viz. :  ,/.  i  t  t 

aj       and       a/ii  + -••  +  «m"m' 

Here  the  as  in  the  second  part  have  the  limit  0  when  h  =  0. 
If  we  replace  a  by  x  and  set  h^  =  dx^,  •••^m  =  ^^mi  1)  becomes 

<^f  =  fr^  (^)  dx^^"-+  /L(^)  ^^m 
=    •-  dx^-\ h  T^  dx^. 

424.    1.  It  is  easy  to  give  examples  of  functions  which  are  not 
totally  differentiahle  at  every  point. 

Ex.  1. 


Consider 

at  the  origin. 
Here 

Hence 


f{x,  y)=V\xy\=  Vxhf 

/i(0,0)  =  0,         /;(0,0)  =  0. 

#=0  (1 

at  the  origin. 

Suppose  now  /  were  totally  differentiahle  at  the  origin.     Then 
the  increment  A/  would,  on  account  of  1),  have  the  form 

A/=aA  +  ^^,  (2 

where  the  limits  of  «  and  /3  are  0. 
This  is  not  possible. 


270  DIFFERENTIATION 

For,  we  have  directly 

A/=/(A,  k}  -/(O,  0)  =  V|M].  (3 

From  2),  3)  we  have 

V^  =ah  +  /3k.  (4 

To  show  now  that  the  limits  of  a,  jS  are  not  0,  let  h,  A;  =  0, 
running  over  the  line  L,  in  the  figure. 
Then 

h  =  p  cos  0,       k  =  p  sin  0.       6  constant. 

This  in  4)  gives 


p  Vsin  d  cos  6  =  p(a  cos  6  -{-  /3  sin  ^), 

or  . 

Vi  sin  2  ^  =  a  cos  ^  +  /S  sin  6.  (5 

If  now 

a  =  0,         y8  =  0, 

the  limit  of  the  right  side  of  5)  is  0 ;  while  the  limit  of  the  left 
side  depends  on  0.     We  are  thus  led  to  a  contradiction. 

2.  Ex.  2. 

fixy)  =  -,  for  a;,  y  not  the  origin. 

^7?  +  ?/2 


0,  for  the  origin, 
lii 

h-- 


Hence 

at  the  origin.     If  now  /  were  totally  differentiable  at  the  origin, 

we  would  have 

A/=  ah  +  I3k, 
or 

r  cos  0  sin  0  =  /•(«  cos  ^  +  /S  sin  ^). 
Hence 

cos  ^  sin  0  =  a  cos  ^  +  /3  sin  0. 

Letting  now  h,  k  =  0,  this  gives,  in  the  limit, 

cos  0  sin  ^  =  0, 
which  is  absurd. 


TOTALLY   DIFFERENTIABLE   FUNCTIONS  271 

425.  Let  f(xy)  be  defined  in  the  domain  D  of  the  point  P  =  (a,  i). 
We  suppose  that: 

a)  f'j.  exists  in  D, 

yS)  fy  exists  at  P, 

7)  f'x  0'^  f'y  i^  continuous  at  P. 
Then  f  is  totally  differentiahle  at  P. 
For, 

=  \f{a  +  h,h  +  k)  -f(a,  b  +  k)l  +  If  (a,  b  +  Jc}  -f^ab)  | 

=  ^i  +  \-  (1 

By  a)  we  can  apply  the  Law  of  the  Mean  to  Aj,  getting 

Aj^  =  hf^(e,  b  +  k}.  a<c<a-\-h  ov  a  +  h<c<a.         (2 

By  ^)  we  have 

^^  =  k\f'y<ia,b^+^U  (3 

where  /8  is  a  function  of  k^  such  that 

lim  ^  =  0.  (4 

From  1),  2),  3)  we  have 

A/=  hf^c,  b  +  k')+  klfy(ab)  +  ^l 
Set 

a=f^(c,b  +  k)-fX(i,b}.  (5 

Then 

A/=  hf^(ab}  +  kfyCa,  b)  +  ah  +  ^k.  (6 

By  7),  lim  a  =  0.  (7 

ft,  *=0 

Equations  6),  4),  7)  show  that/ is  totally  differentiahle  at  P. 

426.  1.   Under  less  general  conditions  we  can  generalize  425 
as  follows  : 

a)  Letf(^x-^  •••  XjjD  be  defined  over  the  domain  D,  of  the  point  x ; 
and  have  finite  partial  derivatives  fl^  •••fl    *^  -^• 

/8)  Let  f'y.    be  a  continuous  function  of  2:^,  x^^-y  •••  x^. 

/c  =  l,  2    ••  m. 

Then  f  is  totally  differ entiable  at  x. 


272  DIFFERENTIATION 

To  fix  the  ideas,  take  m  =  3.     We  have,  setting  for  brevity, 

X-^  =  X^  -f-  11^1       -^2  ^^  "^2  "1"     2'       "^S  ^^^  "^S  "r    '^3  ' 
A/'  =  /(^1^2^3)  -/(^1^2'^3)  =  1/(^1%)  -/(^ia^2^3)l 
+  1/(^1^2^3)  -/(^1^'2^3)1  +  1/(^1%)  -AH^I^Z^I 

By  virtue  of  «)  we  can  apply  the  Law  of  the  Mean  to  each  of 
these  A's,  getting 


^1  =  hAQ^i  +  ^iK  ^v  ^3)' 

(1 

A2  =  h^/LX^v  ^2  +  ^2^2^  ^3)' 

(2 

^3  =  V4(^r  ^2'  ^3  +  ^3^^3)- 

(^ 

Making  use  of  /3),  we  may  write  1),  2),  3) : 

^1  =  ^il/ii(^^1^2^3)  +  Vl' 

(4 

^2  =  ^2/4(^^1^2^3)  +  ^2«2' 

(5 

^3  =  ^3/4C^1^2^3)  +  ^3«3  5 

and  for  A  =  0, 

lim  «j  =  0,     lim  «,  =  0,     lim  «g  =  0. 
Thus 

(6 

Since  the  a's  converge  to  0,  /  is  totally  diiferentiable  at  the 
point  X. 

2.  As  a  corollary  of  1,  we  have  : 

Letf(x-^  ■••  Xjn)  satisfy  the  conditions  «),  /3^  for  a  region  R.     Then 
f  is  totally  differentiahle  at  every  point  of  R. 

427.    1.   Let  the  jjartial  derivative  fljix-^  •■■  Xm)  he.  continuous  in 
the  region  R.      Then  the  difference  quotient 

-r^—  =  -^  umtormly 
Ax^      dx^  ^ 

in  any  limited  perfect  domain  Z),  in  R. 


TOTALLY   DIFFERENTIABLE   FUNCTIONS  273 

For,  by  the  Law  of  the  Mean, 

^=/i.K  -  a^K  +  ^Aa;^  -  O- 

But  /^^,  being  continuous  in  i),  the  function  on  the  right  con- 
verges uniformly  to  /^^,  by  352. 

2.  Let  the  partial  derivatives  of  the  first  order 
he  continuous  in  the  region  R.      Then  the  as  in 

4/"=  if+  fh^^l  +  •••  +  Cim^X^ 

converge  uniformly  to  0,  in  any  limited  perfect  domain  D^  in  R. 
For,  referring  to  the  proof  of  426,  we  have 

Hence  by  1,  the  «'s  are  uniformly  evanescent. 

428.  1.  Let  the  first  partial  derivatives  of  f(x^  •••  a;^)  he  contin- 
uous in  the  region  R.      Then  f  is  continuous  in  R. 

We  have  to  show  that 

lim/(a;i  +  h^  •••  x,„,  +  70  =f(x^  •••  a^m); 

;i=o 

or,  what  is  the  same, 

limA/=0.  (1 

ft=0 

But,  by  426,  2,  /  is  totally  diiferentiable  in  R ;  hence 

A/  =  ^/  +  «i  Ai  +  .  •  •  +  a„Ji^.  (2 

As 

lim  df=  0,         lim  a^  =  0,  •••  lira  a^  =  0 ; 

A=0  A=0  »=0 

passing  to  the  limit  in  2),  we  get  1). 

2.   As  corollary,  we  have  : 

If  all  the  partial  derivatives  of  fix^  •••  x^  of  order  n  are  continuous 
in  the  region  R,  then  f  and  all  its  partial  derivatives  of  order  <.w, 
are  also  continuous  in  R. 


274  DIFFERENTIATION 

429.    1.  At  each  point  of  a  region  R,  let 

A/  =  q^Ax^  +  •  •  •  +  q^^Xm  +  a^Ax^  +  •  •  •  +  a^Ax^ ;  (1 

where  the  qs  are  functions  of  x,  and  the  as  are  functions  of  x  and 

Ax.     Let 

lim«,  =  0.         /c  =  l,  2,  ••.  m.  .  (2 

Then  f  is  totally  differentiable  in  M,  arid 

For,  let  all  the  Aa:/s  be  0  except  Ax^.     Then 

A/ 

Passing  to  the  limit,  we  get  3).     That/  is  totally  differentiable 
follows  now  from  1),  2). 

2.  Let 

df=  j)^dx^  +  ■'■  +  (fin^dx^ 
in  R.     Then 

<^.=  ^-        <^=l,  2,  -m.  (4 


For,  by  definition, 


or 


Let  all  the  dx's,,  except  dx^.,  be  0.     Then 
df=(^^dx^  =  —dx^; 

As  dx^^  0,  we  have  4). 

430.  1.  Letf  =  f(u^'"Un^,andu^  =  gfx^'-'X^^,i  =  l,2,"-n. 
Let  the  image  of  the  region  X  he  the  region  U.  Let  f  he  totally 
differentiable  in  C/,  and  each  g^  he  totally  differentiable  in  X.  Then 
/,  considered  as  a  function  of  the  x's,  is  totally  differentiable  in  X, 


TOTALLY   DIFFERENTIABLE   FUNCTIONS  275 

Since  the  ^'s  are  totally  differentiable, 

A         "^  ^^i  A       ,  -^      A  t=l,  2,  •••w. 

Aw,  =  X  -^  Aa:^  +  Sa  Aa;,.  .,'  (1 

'      -^^  dx^       "       «    «      «  /c=l,  2,  •••w.  ^ 

Since  f(u^  •  •  •  w„)  is  totally  diff erentiable, 

A/=2j^Aw,  +  ?AAw,.  t=l,  2,-w.  (2 

Replacing  the  values  of  Aw,,  given  by  1),  in  2),  we  get 

A/=  fM  fa  +  V  3«3  +  ...  +  V  3^,W 
\3Wj  dx-^      du^  dx^  dUn  dxj 

\du^  dx^      du^  dx^  5w„  dx^J 


where 
when 


Aa^i,  '"  Ax^  =  0. 


(3 


+  7iAa:i+  •••  +7„,A^„; 

lim  7j  =  0,  •••  lim  7^  =  0,  (4 


Then,  by  429, 1),/,  considered  as  a  function  of  the  a;'s,  is  totally 
differentiable ;  and 


Sa?!      5mj  da^i  3w„  5a;j      -^  du^  dx^ 


fit/.      ^T.  ^  ^11,     fl3:„  * 


(5 


bx^      bu^  dx^  aw„  dx„,      ^  du^  dx„ 

^3a;,      '      ^       T^Sw^Sa:,       t1  du^  dx^ 


276  DIFFERENTIATION 

2.  We  have,  /  being  considered  as  a  function  of  the  a;'s, 

or  . 

df=  V  -^  du. 

Thus  to  find  df,  f  considered  as  a  function  of  the  xs^  we  may  first 
find  df  considered  as  a  function  of  the  us,  and  in  the  result,  replace 
the  du's  hy  their  values  in  the  x's. 

3.  As  a  corollary  of  1,  we  have : 
Let 

be  continuous  in  U.     Let 

du^  du^ 

dx. '        dx,r^ 


1  =  1,  2,  •'•  n. 


he  continuous  in  X.      Then  f,  considered  as  a  function  of  the  x's,  is 
totally  differ entiahle  in  X. 

For,  by  426,  2,  /  considered  as  a  function  of  the  m's,  and  the  w's 
considered  as  functions  of  the  a;'s,  are  all  totally  differentiable. 
Hence,  by  1,  /  considered  as  a  function  of  the  re's,  is  totally 
differentiable. 

Some  Properties  of  Differentials.     Higher  Differentials 

431.  In  this  section  we  shall  suppose  the  total  differentials 
which  occur  exist  in  a  certain  region  B,  in  which  x  =  (x^-"X^') 
ranges. 


1.  Let 

Then 

For, 


F=  c-yf-^  +  ...-)-  c„/„.  cs  constants. 

dF=c^df^  +  ...-\-c„df^. 


PROPERTIES   OF   DIFFERENTIALS  277 

2.  Let  F^fg. 

Then 

dF=fdg  +  gdf. 
For, 

dF=V^^dx  =yAdx^+ygfdx^ 

=fX  ^  '^^^ + ^2  %  ^^'  -/^^ + ^'^^• 


3.  Xee 


y/iew 


#  =  •-•        g^OinB. 


dF=^-^l^- 

For,  a^-/-^ 

dF=y'^dx=y-^^^d.^ 

g^dx^     '     g^^dx^     ' 

^ty    fdg  ^gdf-fdg 
9       9"^  9'^ 


432.    The  partial  derivatives  involved,  being  supposed  continu- 
ous in  a  certain  region  ^,  let  us  form  the  expressions 


1|J<K 


"]"~K 


They  are  called  the  second,  third,  ■■■  differentials  oi  f(x-^---Xj^, 
respectively,  in  R. 


278  DIFFERENTIATION 

We  notice : 

since  dx^  acts  as  a  constant  with  respect  to  the  symbol  d. 
Then  1)  gives 

d^f=d-df. 

In  the  same  way  we  find 

(i»/=  d  •  (^«-y=  d^  •  (^"-2/ ... 

433.  1.  Let  w=f(u^"'Uj,^,  while  u^-'-u^  are  functions  of 
2^1  '•  •  ^»j.  Let  w,  when  considered  as  a  function  of  the  x's,  be 
denoted  by  ^(a;j  •••x^).  Let  finally  all  derivatives  involved  be 
continuous.     Then 

dF^^'£du,.  (1 

^F^^dn^d.'^  +  X'ia^. 


PROPERTIES   OF   DIFFERENTIALS 


279 


d^F=  V  du^  du,  d  •  — ^  +  y  — ^  du  d?n 


t    5w^ 


=s 


5?f 


32/- 


5/ 


^  du^  du^  du^ 


du^  du^  du.  4-  3  "V  - — "- —  d^u^  du^  +  ^  tA  d^u^ 


7'3m, 


(3 


l,/C  t   "        K 


In  the  same  way  the  higher  differentials  d^F—  can  be  calculated. 

2.    Incase  ^u,  =  0,  d\=0,  -. 

we  have  ^^ ^  ^^^  ^^^  ^^^  ^3^  ^  ^3^^  ... 


434.    Let  all  the  partial  derivatives  of  f(x-^  '"^m)  ^f  order  n  he 
continuous  in  the  domain  D  of  the  point  x. 
Then^  if  x  +  h  lies  in  D, 

f(x^  +h^---x,^  +  h^}  =  f(x^  ■  •  ■  ^'"^  +  JT  ^^^^1  '■'  ^'"^  "•"  2]  ^-^^^^ '"  ^'"^ 


setting 


The  expression  on  the  right  of  1)  is 
called  Taylor  s  development  of  f  in  finite 
form. 

For  simplicity  we  shall  suppose  7n  =  2. 
The  reasoning  in  the  general  case  is 
precisely  the  same. 

To  avoid  writing  indices,  we  shall  call 
the  variables  x,  y. 


(1 


280  DIFFERENTIATION 

Let  X,  y  be  any  point  on  the  line  L  joining  the  points  a,  h  and 
a  +  h,  h  -{■  k.     Then 

x=a  +  uh,    y  =  h  +  uk.         0  <u<l. 

Also  let 

f(x,  y)  =  f{a  +  uh,    b  +  uk)  =  g(u). 

Then,  when  u  runs  over  the  interval  %  =  (0,  1),  the  point  xy 
vnns  over  the  interval  on  L  between  a,  b  and  a  +  h,  b  +  k. 
By  433,  1,  we  have 

y'(u)A  +  ^£k=df(x,y-), 

g"(u)  =  dy(x,  y},   -.  g^^Ku-)=d^f(x,  y). 

Thus  g(u)  and  its  first  n  derivatives  are  continuous  functions 
of  u  in  5t- 

Applying  409,  we  have 

K^)  =  KO)+^y(0)  +  ^y'(0)+.-+^^(")(^u). 

0<^<1. 
Setting  here  u=l,  and  observing  that 

^(l)  =  /(a  +  A,  5  4-A;),  ^(0)=/(a,J), 
/(0)=  df(a,  5),  y  (0)=  £?y(a,  5),  ... 
g("X0u}  =  dZf(a  +  (9A,  b  +  <9^), 


we  get 


/(a  +  A,  J  +  A:)  =  /(a,  5)  +  i^  (^/(a,  5)  +  i^  (^/(a,  5)  + 


+  -\r  d-'f(a,  5)  +  i-  (^«/(a  +  ^A,  J  +  Ok) . 
n—ll  nl 


435.  In  Taylor's  development  of  a  function  /(a;)  of  a  single 
variable  [409],  we  have  only  assumed  that  f^''\x)  is  finite  within 
% ;  whereas,  in  the  corresponding  development  of  a  function  of 


PROPERTIES   OF   DIFFERENTIALS  281 

several  variables,  we  have  assumed  [43J:]  that  all  the  partial 
derivatives  of  order  n  are  continuous  in  i>(a),  in  order  to  use  430. 

It  is  interesting  to  note  that  the  development  may  not  hold  if 
these  derivatives  are  not  continuous. 

Consider  the  function 

f(xy^  =  ^\xy\, 
employed  in  424,  1. 

We  have 

•^•^      '/»  • /I Tq'    Jy~9     /j To'  x.y^yj. 

/i(:r,  0)  =  0,  fl(0.y)=0. 

The  derivatives  of  the  first  order  are  thus  continuous,  except  at 
the  origin. 

Let  P  =  (x,  a;),  Q  =  (x  -^  h,  x  +  h)  be  two  points  on  the  line 
y  =  x,  which  we  call  L. 

If  now  Taylor's  development  were  true  in  a  domain  about  a,  in 
which  the  nth.  partial  derivatives  were  finite,  we  could  write,  tak- 
ing here  7i  =  1, 

/(^  +  7^,  :,  +  A)  =/(:r,  x)  +  h\fX^.  0+fyil  OK  (1 

where  (|,  |)  is  a  point  on  L  between  P,  Q. 

This  formula  should  be  valid  for  all  x,  h.  But  in  the  present 
case 

Thus  1)  gives 

\x+'h\  =  \x\  +  h'&gn  ^.  (2 


That  this  result  is  false  is  easily  seen. 

For  example,  let 

a;=  —  1,    A  =  5. 

Then  2)  gives 

-^ 

4  =  1±5,        ifl^O 

^^ 

Y  ^ 

=  1,               if  1=0. 

Co    <^ 

OJ? 

•V^ 

o 

^ 

CHAPTER   IX 
IMPLICIT  FUNCTIONS 


436.   1.   Let 


be  a  relation  between  the  m  -f  1  variables  x^^  •"  Xj^,  u.     Let 

a^j  =  ^j,  •  •  •  Xjj^  =  a^ 
be  a  set  of  values  such  that  the  equation 

F(a^-a,,,u)  =  0  (2 

is  satisfied  for  at  least  one  value  of  u ;  i.e.  the  equation  2)  in  u 
admits  at  least  one  root.  Let  D  be  the  aggregate  of  the  points 
x  =  (x^  -••  Xjn)  for  which  1)  has  at  least  one  root  u.  We  may  con- 
sider M  as  a  function  of  the  x's,  u=  (f>(x-^^  •••  x^}  defined  over  D, 
where  </>(a:j  •••  x^)  has  assigned  to  it  at  the  point  x,  the  roots  u  of 
1)  at  this  point. 

We  say  u  is  the  implicit  function  defined  by  1).    It  is  in  general 
a  many  valued  function. 

EXAMPLES 

1.  Let 

y=f(x)  (3 

be  defined  over  a  domain  D.    Let  E  be  the  image  of  D.    Then  3)  defines  an  inverse 

function  ,  . 

x  =  g(y), 

defined  over  E,  by  217.  This  same  function  may  be  considered  as  an  implicit 
function,  defined  by  ^/  \      t^^       \     a 

2.  Let  /(«)  =  1  for  every  sc  in  Z)  =  (01).     If  we  set 

the  image  E  of  Dis  the  single  point  y  =  1.     The  inverse  function 

x  =  g(y), 
is  defined  only  for  ?/  =  1  ;  at  this  point  g  takes  on  all  values  between  0  and  1. 

282 


IMPLICIT   FUNCTIONS  283 

3.   Let  i^  =  0  be  the  relation 

x2  +  y2  +  a2-r2  =  0,        r=^0.  (4 

At  each  point  of  the  domain  D, 

a:2  +  2/2^r2, 

the  equation  4)  admits  one,  and  in  general,  two  values  of  z.  The  equation  4) 
therefore  defines  z  as  a  two-valued  implicit  function  u  of  x,  y,  over  the  domain  D. 

*■  x2  +  2/2  +  ^2  =  0.  (5 

In  this  case  there  is  only  one  set  of  values,  viz.  x  =  y  =  z  =  0  satisfying  5).    Thus 
z  is  defined  only  for  a  single  point,  viz.  x  =  y  =  0.     At  this  point,  2  =  0. 

^'  x2  +  2/2  +  22  +  r2  =  0.        r^  0.  (6 

This  equation  is  satisfied  for  no  set  of  values  of  x,  y,  z.     The  equation  6),  there- 
fore, does  not  define  any  function  z  of  x,  y. 

6.  sin2  u  +  cos2  it  —  =  0.  (7 

y 

This  equation  admits  no  solution  except  for  points  on  the  line 

y  =  x. 

For  all  points  on  this  line,  the  origin  excepted,  the  equation  7)  is  satisfied  for 
any  value  of  m  in  'St. 

2.  More  generally,  let 

('S^ 

be  a  system  of  p  relations  between  the  m  +p  variables  a;,  u.  Let 
D  be  the  aggregate  of  points  a:  =  (a:j  •••  a;^),  for  which  the  system 
)S  is  satisfied  for  at  least  one  set  of  values  of  u^--'Up.  We  may 
consider  the  w's  as  functions  of  the  x's. 

where  the  ^'s  have  assigned  to  them  at  the  point  x,  the  values  of 
the  roots  u■^^^■^  u^  at  this  point.  We  say  u^--  Up  is  a  system  of  im- 
plicit functions  defined  by  the  system  S.  These  functions  are,  in 
general,  many  valued. 


284  IMPLICIT   FUNCTIONS 

3.   Suppose  we  know  that  a  set  of  values 

X-^  =  ftp  •••  X„^  =  d^,  Wj  =  Oj,  •••  Up  =  Op 

satisfies  the  system  S.  Let  us  call  the  set  of  values  u^  =  b^---Up  =  hp 
initial  values. 

We  wish  to  show  now  that  under  certain  conditions,  the  system 
S  defines  over  a  region  M  a  set  of  p  one-valued  continuous  func- 
tions u^---  Up  in  the  variables  x^---  a:^,  satisfying  6'  for  every  point 
of  M,  and  taking  on  the  above  initial  values  at  the  point  x=  a. 
Furthermore  there  is  only  one  such  system  of  functions. 

The  method  employed  is  due  to  Goursat,  Bull.  Sac.  Math,  de 
France.,  vol.  31  (1903),  p.  184.  It  rests  on  a  principle,  having 
many  applications  in  analysis,  known  as  the  Method  of  Successive 
Approximation. 

437.  1.  Let  us  first  consider  only  two  variables.  Tlie  method 
employed  for  this  simple  case  is  readily  extended  to  the  most  gen- 
eral case.     We  begin  by  establishing  the  fundamental 

Lemma.  LetfQx.,  w)  he  continuous.,  and  -~  exist  in  the  domain  J), 
defined  hy 

21;  |a;— a|<cr, 

4B;  \u-h\<r. 

Let  f  vanish  at  a,  h.  Let  6  be  an  arbitrary  positive  number  <  1, 
such  that 

f<0,  inB,  (1 

du 

while 

|/(:r,  6)|<t(1-^)=77<t,  m  21.  (2 

Then 

u-b  =/(:c,  u)  (3 

admits  one  and  only  one  solution 

U=(f>(x),         in%. 

which  is  continuous  at  a,  and  takes  on  the  initial  value  u  —  b  at  x  —  a. 


IMPLICIT   FUNCTIONS  285 

The  function  (j)  is  continuous  in  21,  and  remains  in  ^  while  x  runs 
over  31. 

We  set 

Wj  -  5  =/(«,  ^),  u^-h  =  f(x,  u{),  u^  -  5  =  f(x,  ^2),  ••• 

Then  all  these  us  fall  in  ^. 

For,  by  2),  Wj  falls  in  ^.     Let  us  admit  that  u^-i  falls  in  :33»  and 
shoAv  that  %  also  falls  in  ^. 

In  fact,  by  the  Law  of  the  Mean, 

Uj.—  u^  =  (Uy  —  5)  —  (mj  —  5)  (4 

Hence,  by  1), 

I  u,.  —  u-^\<6\  u^-i  —  b\  (5 

<0T,  (6 

since,  by  hypothesis,  Uj._i  falls  in  ^. 

Thus,  from  u,.—  b  =  (u^  —  %j)  +  (?*j  —  h}  and  6),  we  have 

\Uj.  —  b\<c\u^  —  b\  +  dr. 
But 

l2*i-6|  =  |/(:r,6)|<(l-^)T,  by  2). 
Hence, 

\Ur-b\<T;  (7 

I.e.  all  the  us  fall  in  ^. 

We  show  now  that  for  each  x  in  21, 

U=  lim  w„=  (^(a;) 

is  finite.     To  this  end  we  show 

e  >  0,     m,     I  w„  —  w^  I  <  e,     n>m.  (8 

For,  in  the  same  way  that  we  established  5),  we  can  show  that 

\Uj.—  Uf.^l\<d\Ur^l—Ur-2\'  (^ 


Hence 


286  IMPLICIT   FUNCTIONS 

Thus,  we  get 

\u^  — Ui\<0\u-^  —  b   \<dr), 

\u^  — u^\<d\u^  — u-^^\<6^r},  (10 

I W4  —  ?^3 1  <  ^  I  Wg  —  ^2  i  <  ^^Vf  6tc. 

K-Um\<V0%'^  +  (9+  ...  +  0n-m-l-^ 

<-^,         since  0<^<1 

1  —  u 

if  m  is  taken  sufficiently  large. 

Thus  the  relation  8)  is  established. 

Furthermore,  the  above  reasoning  shows  that  one  and  the  same 
m  suffices  wherever  x  is  taken  in  31.  Thus  m„  converges  uniformly 
to  U  in  St. 

Finally,  by  virtue  of  7),  C/"  falls  in  ^. 

The  function  U  satisfies  3)  in  21. 

For,  in 

let  n=  cx>.     Since  /  is  continuous,  we  get  in  the  limit 

U-b=f(_x,  U). 

We  show  now  that  Z7=  ^(a;)  is  continuous  in  5t.  For,  since  w„ 
converges  uniformly  to  <j>(x}  in  21,  we  have 

cf>(ix  +  h')=uXx-{-h')-\-€\  W\<1- 

if  n  is  taken  large  enough. 

But  Wj,  u^,  ■■•  are  continuous  functions  of  x,  since /  is  continuous. 
Thus,  for  sufficiently  small  S, 

|Wn(2;  +  A)-w„(a;)|<| 


for  I  Al  <  S  and  x  +  h  in  21- 


IMPLICIT   FUNCTIONS  287 

Hgiicg 

\<f>(x+h)-(f>(^x)\<e. 

We  show  now  that  U  is  the  only  root  of  3)  which  is  continuous 
at  a  and  takes  on  the  initial  value  5  at  a. 
For,  let  V=  'i^i^x)  be  such  a  solution. 
If  X  is  taken  sufficiently  near  a,  V  falls  in  ^. 
Then  from 

w„-6=/(a:,  w„_i), 
we  have,  by  the  Law  of  the  Mean, 

\V-u,\  =  \Kx,  F)-/(a;,w„_0| 


du 


<d\V-u,_i\,     byl), 

Hence,  passing  to  the  limit,  n  =  oo, 

V-U  =  0,    for  all  points  of  51. 
2.  As  corollary  of  1,  we  have : 

be  continuous  in  the  domain  of  the  point  (a,  5),  and  vanish  at  that 
point. 

Th^^  u-b=f(ix,u-) 

admits  a  unique  solution 

u  =  (l>(x'), 

which  is  continuous  in  the  domain  of  x=  a,  and  has  the  initial  value 
u  =  h  at  X  =^  a. 


288 


IMPLICIT   FUNCTIONS 


438.    1.   By  means  of  the  preceding  lemma,  we  can  now  prove 
the  theorem  : 

Let  F(^x,  u)  be  continuous,  and  F'^  exist  in  the  domain  2),  defined  by 

21 ;  \x—  a\<a, 

«;  \u-b\<T. 

At  a,  h  let  F=  0,  while  Fl  ^  0. 

Let  6  be  an  arbitrary  positive  number  <  1,  such  that 


ivhile 


F'uia,  b)\ 


F(x,  b) 


<(1-^)t, 


in  21. 


Then  the  equation 


(1 
(2 
(3 


FL(a,  b) 

F(x,  m)=0 

is  satisfied  by  a  one-valued  continuous  function 

u  =  <^(x)i         in  21; 


having  the  initial  value  b  at  x=  a,  and  which  remains  in  ^  while  x 
is  in  2t. 

Furthermore^  3)  admits  no  other  solution  which  is  continuous  at  a, 
and  has  the  initial  value  b  at  a. 


For,  consider  the  equation 


—  b  =  u—b  —  -—, — ^^—^  =  fix,  u)' 
F'uia^b)     ^^  '    ^ 


(4 


Evidently  this  is  equivalent  to  3);   i.e.  every  function  u  which 
satisfies  3)  satisfies  4),  and  conversely. 
Here 

df  ^  ^      Fljx,  u-) 

du  Flia,  b) 

Hence  /  is  continuous,  and  fl  exists  in  L;    also  /  vanishes 
at  a,  h. 


IMPLICIT   FUNCTIONS  289 

Furthermore,  from  1) 

!^!<^,         in  I). 
\du\ 

while  from  2), 

|/(:r,  6)|<(l-^)r,  in  51. 

Thus  /  and  /^  satisfy  all  the  conditions  of  the  lemma  in  437, 
and  the  theorem  follows  at  once. 

2.  The  reader  should  remark  that  the  preceding  theorem  makes 
no  assumption  regarding  F'^.     This  may  not  even  exist. 
For  example,  let 

a;sin-=0,  fora^=0. 

Consider  .. 

F(x,  u)  =  u^  —  X  sin  -  =  0.  (5 

X 

Here  F'^  does  not  exist  at  a;  =  0,  w  =  0.  However,  the  equation 
5)  defines  a  continuous  one-valued  function,  which  takes  on  the 
initial  value  w  =  0  for  x=0  ;   viz.. 


3/        .      1 

=  \/a;sin  — 


3.  As  corollary  of  1  we  have  : 
In  the  domain  of  the  point  a,  J,  let 

F(x,  w),  F'u(x,  u) 

be  continuous.     At  the  point  a,  b,  let 

F=0,  F[,=^0. 

TJien  the  equation  F(x^  w)  =  0  admits  a  unique  solution 

u  =  <^(a;), 

which  is  continuous  in  the  domain  of  the  point  x=  a,  and  has  the 
initial  value  u  =  b,  at  this  point. 


290  IMPLICIT   FUNCTIONS 

439.    We  have  seen  in  438  that 

Fix,u)=0  (1 

is  satisfied  by  a  continuous  function 

u  =  (^i(a;) 

in  a  certain  interval  {a  —  a,  a  +  o-)  =  (vl,  5);  and  that  there  is 
only  one  such  function  which  =  b,  when  x  =  a.  In  general  this  is 
true  not  only  for  the  interval  {A,  B^  determined  by  the  theorem 
438,  but  for  a  larger  interval  (C,  D),  containing  (^,  B^.  For,  let 
flj  be  a  point  near  one  of  the  end  points  of  (A,  B^.  Let  6j  =  ^^(aj). 
Let  us  replace  a,  b  in  the  theorem  of  438  by  a^,  5j.  Then  the  con- 
ditions of  this  theorem  are  satisfied  for  a  certain  interval  (-4^,  -Sj), 
about  aj ;  to  which  corresponds  a  continuous  function 

u  =  (f)^(x'), 

determined  by  the  condition  that  u  =  5j,  for  x  =  a^.  The  interval 
(^j,  5j)  will  in  general  extend  beyond  (A,  B}.  In  the  interval 
{Ay  B)  which  the  two  intervals  {A,  B),  {A^,  B^  have  in  common, 
the  two  functions 

are  equal.     Let  us  define  a  function 

u  =  4>(x)  =  4>^(x),  in  {A,  B), 

=  <^2(^)'  ill  (^1'  A)- 

Then  the  equation  1)  is  satisfied  by  this  function  in  {A,  B{)^ 
and  it  is  uniquely  determined  by  the 
fact  that  it  is  continuous  in  (^,  J?j) 
and  has  the  value  u  =  b,  for  x=  a. 
In  this  way  we  can  continue  extend- 
ing on  the  right,  and  on  the  left,  the 
original  interval,  until  we  are  blocked 

by    certain    points   beyond    which    we   

cannot    go.       Such    points    may   arise  laa^B    i 

when  F(x,  u)  ceases  to  be  continuous,  or  when  FJ  =  0. 


IMPLICIT   FUJsCTIONS  291 

440.    1.  We   proceed   now   to   extend   the  theorem  of   438  to 
embrace  the  system  S  of  436. 

To  this  end  we  generalize  the  lemma  of  437  as  follows : 
Lemma.     Let 

fiC^i  ■  • '  ^m^i  •-Up)---  fp(x^  ■  ■  ■  x^u^  •  •  •  Up), 
and 

i         ,,«  =  !,  2,  ...;,. 

be  continuous  in  the  domain  D  defined  by 

21;  \^\- a^\<(T  ■■■\x^  —  a„,\<a, 

® ;  \u-^  —  b-y\<r---\Up—bp\<T, 

and  let  f.^,  •■■  fp  vanish  at  the  point  (a^  •■•  a,„5j  •••  5^). 
Let  0  be  an  arbitrary  positive  number  <  1,  such  that 


5/ 


du^ 


Q 

<  — ,         t,  /c  =  1,  2,  •••  »,     in  D.  (1 

P 


while 

\f,(ix^-xJ^-b;)\<T(l-e)='n,     in%.       t=l,  2,  ...;>.       (2 
fPhen  the  equations 

admit  one,  and  only  one,  set  of  solutions 

Ux  =  (f)xCx^---  x,,,)      •■■      Up=  4)p(^x^ ' ' •  rr^) 

in  51,  which  are  continuous  at  a,  and  take  on  the  initial  values  5 j  •  •  •  b^, 
at  x=  a. 

The  functions  (p  are  continuous  in  3t,  and  remain  in  SS,  as  x  runs 
over  31. 


We  set 


Upi  -bp  =  fpQx^  •  •  •  xj-^^  •-•bp) 

^P2        ^P  ~/pC-^l  ■■■  ■^m'^ll  "°  '^plJ 

etc. 


292  IMPLICIT  FUNCTIONS 

We  show  now  that  all  these  us  lie  in  Sd. 

For,  by  2),  u^^---  Up^  are  in  ^.  Let  us  assume  now  that 
Ui^r-i"-  ''^p,r-i  lie  ill  -^'  and  show  that  u^^---Up^  also  lie  in  ^.  By 
the  Law  of  the  Mean, 

the  arguments  of  these  derivatives  lying  in  D.     Hence,  by  1), 

n 

\u^r-Ua\<  —  \\Ui^r-l  —  ^l\-i \-\Up,r-l  —  f>p\l  (^ 

Thus,  as 
we  have 

Kir  ~  ^i|<  |*^U  —  ^J  +  ^''■7  t  =  1,   2,   •••p. 

or  using  2), 

l^tr— M<Ti      4=  1,  2,  •••jt?;  r=2,  3,  ... 

which  was  to  be  shown. 

We  show  now  that  for  each  x  in  21, 

U^  =  lim  u^^  ^  =  (f)^(x^...  x^') 

is  finite. 

To  this  end  we  show  that 

e>0,      m,      \u^n  —  u^^\<€.         n>m.  (4 

1  =  1,2, -p. 
For,  as  in  3), 

r 

\'^i2  —'^ui\<^Vi     by  2)  and  3), 
\u,a  -u,^2\<^\   by  5), 

\'^i,i—  '^i,3\<  ^^V-<  6tC. 

These  relations  are  analogous  to  the  relations  10)  in  437.  The 
rest  of  the  demonstration  can  now  be  conducted  as  in  437  to  estab- 


IMPLICIT   FUNCTIONS  293 

lish  not  only  the  relation  4),  but  the  remainder  of  the  theorem  in 
hand. 

2.   We  can  state  1  in  a  form  less  explicit,  but  easier  to  remem- 
ber, as  follows : 

Let 
and  „  „ 

he  continuous  in  the  domain  of  the  point  a^  •••  amh-^  •••hp. 
Let  these  p^  -\-  p  functions  vanish  at  this  point. 
Then  the  system  of  equations 

u^-h^  =  f^(x^---Up)      •••      Up-lp=fj,(x^--'Up) 

admits  a  unique  system  of  solutions 

which  is  continuowi  in  the  domain  of  the  point  a:j  =  a^  •••  a;^  =  «^, 
and  takes  on  the  initial  set  of  values  u^  =  b^--- 11^=  b^. 

441.    We  can  now  generalize  438  as  follows : 

Let 

Fi(x^---x,„7i^---Up')      ■■■     Fp(x^---x„,u^---Up'), 

and 

dF 

^        .,.  =  1,2,  ...p.  (1 

be  continuous  in  the  domain  D  of  the  point 

V  5  •''1  ^^  ''l  "  '  ^m  ^^  ^mi    U\  =  O^"-  Up  =  Up. 

Let  F^---Fp  vanish  at  Q,  while  the  derivatives  1)  have  the  values 

d^         at  Q. 


Let 

\d     .-.d     I 
i"ll       "l?^ 


I  ^p\  ' ' '  ^pp 


^Q. 


294  IMPLICIT   FUNCTIONS 

Then  the  system  of  equations 

18  satisfied  hy  a  set  of  functions 

which  are  one-valued  and  continuous  in  a  certain  region  31,  about  the 
point 

a  \  x^  =  a-^"- Xjf^  =  aj^ ; 

and  at  this  point,  these  functions  have  the  values 

u^  =  b^---Up=  bp. 

Furthermore,  the  system  S  admits  no  other  set  of  p  functions,  con- 
tinuous at  a  and  taking  on  the  initial  values  b  at  that  point. 

We  replace  the  system  /S  by  the  equivalent  system 

^ii(«*i  -  ^i)  +  •••  +  d^(Up  -  bp)  =  d^^Qui  -bj)-\-'" 
+  d^p(Up-bp)-F^=gj_ 


(2 


+  dpp(Up  -  bp)  -Fp  =  gp. 

Since  A  t^  0,  we  can  solve  this  system  for  the  differences  Ui  —  bi, 
and  get 

Wj  _  5j  =  e^^g^  +  •  •  •  +  e^pgp  =f^ 

(3 

Up- bp  =  epig^  + ■■■-{-  eppgp  =fp. 

Obviously,  the  functions  g,  and  hence  the  functions  /,  are  con- 
tinuous in  D. 


IMPLICir   FUNCTIONS  295 

So  are  the  derivatives  r^-     For 

where 

Since  the  ^'s  vanish  at  Q,  so  do  the  /'s.  Since  the  derivatives 
5)  vanish  at  Q,  so  do  also  the  derivatives  4). 

Obviously,  therefore,  the  numbers  a,  t,  0  of  lemma  440  exist,  such 
that 


3/ 


<  -,         in  D. 


We  can  therefore  apply  this  lemma  to  the  system  3).  Since  this 
system  and  the  given  system  )S  are  equivalent,  the  theorem  is 
proved. 

442.    1.  Let  f(xi-x^,u)=0  (1 

admit  a  solution  u  =  b^  at  the  point  x  =  a.  In  D(a^  5),  let  f(xi  •  •  •  x^^u) 
have  continuous  first  partial  derivatives.  Let  fl^^  in  D.  Then 
1)  defines  a.  one-valued  fu7iction  u,  in  a  certain  domain  A,  of  the  point 
a,  whose  first  partial  derivatives  in  A  are  given  hy 

^  =  -^.        .  =  1,2,  ...r^i.  (2 

For,  let  a;  be  a  point  of  A.  Let  x  receive  the  increment  Aa:t, 
while  the  other  coordinates  of  x  remain  constant.  Let  the  corre- 
sponding increment  of  u  be  Aw.     Then 

f(x^---xi  + ^Xf-Xj^u  +  /:i.u')—f(x-^--'X^u')=0,  (3 

by  virtue  of  1).  Applying  the  Law  of  thfe  Mean  to  3),  we  have, 
setting  x[  =  Xi-\-  6Axi,  u'  =  u  +  dAu, 

f'^Jix^ •■■x[--- xy^Ax,  +f'uix^---x[--- x^u'}Au  =  0  ; 


296 
whence 


IMPLICIT   FUNCTIONS 


Passing  to  the  limit,  we  get  2). 

2. 

For,  by  2), 


df=^^dx,+...  +  ^Jx,^  +  fju  =  0. 


du 


^0.        (4 


du  =  — T^  dx^ 
du 


dx^ 
du 


dx^. 


Multiplying  by  the  common  denominator,  we  have  4). 

443.    1.  Let  the  system 

F^(x^---x„,,Uj^---Up')=0 

admit  a  solution  u  =  b  at  the  point  x=  a.     Let  the  functions  F-^---Fp 
have  continuous  first  partial  derivatives  in  L(^a,  6).     Let 


(1 


J= 


5mj      du^ 

dl\     dF, 

BUp      du^ 


=^0, 


in  D, 


Then  1)  defines  a  system  u-^-'-u^  of  one-valued  functions  in  a  cer- 
tain domain  A  of  the  point  a,  whose  first  partial  derivatives  — '  in  A 
are  given  by  the  system  of  equations  " 


ai\_^5^awi_^_^3^5^^^ 


dx 


du,  dx, 


dUp  dx^ 


(2 


dFp_^^J\,du,^_^d_Fpd_Up^^ 
dx^        5?<j  dx^  dUp  dx^         ' 


with  non-vanishing  determinant  J. 


IMPLICIT   FUNCTIONS 


297 


For,  let  P  =  (x^- •  ■  a;^Wj  •  •  •  u^  be  a  point  of  B.  Let  Awj  •  •  •  Awp  be 
the  increments  of  u^---Up,  corresponding  to  an  increment  Ax^  of  x^. 
Let  0  <  ^j  <  1,  and 

Qc  =  (xj^---x^  +  O^Ax^'--x„„  u^  +  6 ^Au^--- Up  +  O^Aup).    t  =  l,  2,  '--p. 

Let  <^t  be  the  value  of  -— ^,  and  yjr^^  be  the  value  of  - —  at  Q^. 
Then,  by  the  Law  of  the  Mean,  we  have  from  1), 


A^i  =(/>!  + til  l^  +  ^i: 


Au^ 

'  Ax^ 


+  -  +  t 


Am, 


ip 


A?/i 

Ax^ 


AFp  =  ct>p  +  y{rp,^  +  y}rp,^  +  -  +  fpp^^ 


A  1^2 

Ax^ 


Ax. 


Am, 


^=0 


(3 


0. 


Thus, 


AUl 

Ax^ 


■^11  •••^IP 

"^21  •••^2P 
i^Pl---fpp 


Let  Aa;^  =  0;  the  limit  of  the  right  side  exists,  since  the  partial 
derivatives  of  the  F's  ave  continuous,  and  J^O.         Hence  the 

derivatives   — ^   exist.     Hence  in   the  limit,  the  system  3)  goes 

over  into  tlie  system  2). 

2.  The  determinant  J  is  called  the  Jacobiati  of  the  system  1). 


CHAPTER   X 
INDETERMINATE  FORMS 

Application  of  Taylor  s  Development  in  Finite  Form 

444.  The  object  of  the  present  chapter  is  to  show  how  in  cer- 
tain cases  we  may  determine  the  limit  of  expressions  of  the  type 

which,  on  replacing /(a;),  g(x)  by  their  limits,  assume  the  forms 

— ,      — ,      0  •   GO,      GO  —  GO,      1°°,      0'^,      GO**. 
Q'      QO'  '  1  1  •> 

These  are  ordinarily  called  indeterminate  forms. 

445.  Suppose  by  the  aid  of  Taylor's  development  in  finite  form, 
or  otherwise,  we  find  that,  in  R  =  RD(a)^ 

f(x)=  a(x  —  a')"^  +  (f)(x)(^x  —  a)"*,         m'  >m. 

g(x)  =  ^(x  —  ay  +  '^(x) (x  —  a)^\  n'  >  n. 

where  <^,  yjr  are  limited  in  M,  and  a,  ^4^0. 

g(x)      /3  +  (a;  -  a)'^ -"i/r    ^  ^ 

Passing  to  the  limit  a;  =  a,  we  have 

0,         if  w  >  w. 


i21im4^  = 
«=«  g(x) 


a//3,    if  m  =  n, 

(T  •  cc,  II  m<n.         cr  =  sgn  — 


Similar  considerations  apply  to  the  left  hand  limit  at  a. 

298 


TAYLOR'S   DJ:VEL0PMENT  IN   FINITE   FORM  299 


gi gsiiii 

Example.  1™ -. —  =  +  1. 

^  x=o  £c  —  Sin  X 


For,  „       _2       ~3 

1!2!3!         ^'■^  ^ 


/(«)  =  e^  -  e"°^  =  ^  x3  +  a;V(5c). 
gr(x)  =  X  —  sin  a;  =  ^  x^  +  x^\p{x). 


Similarly, 


The  functions  <(>,  f  are  limited  in  Z>(0). 
Thus, 


fW      I  +  x<t>(x') 


g{x)      \^-x^{xy 
whose  limit  for  x  =  0  is  1. 


446.    To  find  the  limit  of 

Ax^-gix-),  (1 

when  /  and  g  are  infinite  in  the  limit,  we  may  sometimes  find  a 
development  of  /(a;),  gix)  in  the  form 

0^  +  (. -Tr-  +  ■  ■  +  "0  +  Hi-  --»)+■■■+  (^  -  ay^i^-), 

valid  in  -Z)(a)  or  RD(^a).,  the  function  <^  being  limited  here. 

This  method  of  finding  the  limit  of  1)  is  best  illustrated  by 
an  example. 

lim  ( cosec  x )  =  0. 

1=0   \x  / 

We  have 

cosec  x= = ,       rf)Cx)=i. 

sinx     x{l  -x2</)(x)}  ^         ® 

l-x^<t>{x)  l-x2,/,(x)         ^     "^^  ^'        i^w     ? 

Hence 

cosec  X  =  -  {1  +  x2^  (x)}. 

Therefore 

—  cosec  x  =  —  xf(x)  =  0. 


300  INDETERMINATE   FORMS 

447.    When  the  independent  variable  a;  =  +  oo,  we  may  set 

1 

u 

which  converts  the  limit  into  B,  lim,  by  290. 

«=0 

I 
Example.  y  =  x{a^  —  1 )  t        a  >  0. 

lim  y  =  log  a. 

X=+oo 

For,  ■  1 


where 
Hence 


0X  -qU  -  e«  loga  _  1  +  j^  log  Qj  ^.  u'^^(jt), 

(p(u)  =  i  log'^a. 

y  =  log  a  +  U(p(u) 
=  log  a. 

448.  When  the  preceding  methods  are  not  convenient,  we  may 
often  apply  with  success  one  of  the  following  theorems.  These 
rest  on 

Cauchijs   theorem.     Let  f(x),  ,9'(^)  ^^    continuous   in   21  =  (a,  5). 

Within  9t,  let  f  (x)  he  finite  or  infinite  and  g' (x)  finite  and  ^0, 

Then 

f(h-)-fia^      f\c-) 


g(b)-g<ia)      g'Qc) 


a<e<h.  (1 


We  note  first  that  g{h^^g(^a).  For,  if  g(b^=g(^a^,  we  can 
apply  Rolle's  theorem  to  g{x),  which  shows  that  ^'(a;)  must  vanish 
within  21,  which  is  contrary  to  the  hypothesis. 

To  prove  1),  we  introduce  the  auxiliar}'^  function 

Obviously,  h(x^  is  continuous  in  2t.  Also  for  points  within  21, 
for  which  /'  (a;)  is  finite, 

while  for  the  other  points  within  2t,  h'Qx^  is  definitely  infinite. 
Finally,  we  observe  that  A(«)  =  A(J)  =  0.  We  can  thus  apply 
Rolle's  theorem  to  ^(a:;),  which  gives  1)  at  once. 


THE   FORM  0/0  301 

449.  1.  Let  Ax-),  fix)  -/"-^a;),  g(x).  /(a:)  ■■■  g^^-^Kx)  he 
continuous  in  21=  (a,  a  +  8),  and  vanish  for  x  =  a.  Letf-^^x)  be 
finite  or  infinite  within  21.  Let  g^^^x)  he  finite  and  ^  0  within  %. 
Let  g'(x),  g"ix')  ■  ■  ■  g^^'^Xx)  ^  0  within  21.      Then 


For,  by  448, 


f(a  +  h)      f^\c)  ,   ,                    ,. 

•^ — — r^  =  -^,  ,;  (  •  a<o<a  +  h.                  (1 
gia  +  h)      g<-\c) 

Ka+h)_fie{)  ^.,.^.^. 


a  <  Co  <  <?i,      etc. 


2.   We  note  that  the  denommator  ^(a  +  A)  in  1)  is  ^fcO.     For 
otherwise,  g'  (jc)  would  vanish  somewhere  within  21- 


The  Form  5 

450.  1,  Let  /(a;),  g(x)  he  continuous  in  R  =  RD(a),  and  vanish 
at  a.  Let  g'ix)  he  finite^  and  t^O  within  R.  Let  f  (x)  he  finite 
or  infinite  within  R.     Let 

R  lim     ,  ,  i  =  X,         iinite  or  infinite.         (1 

where  x  runs  over  only  those  values  for  which  f  (x)  is  finite. 

Then  j.^  . 

i21imiZW  =  X.  (2 

x=„  g{x) 


For,  by  449, 


a  <  ^  <  X. 


The  limit  of  the  right  side,  as  x  =  a,  is  X. 
Hence  the  limit  *  of  the  left  side  is  X,  for  x—  a. 

*  The  reader  should  bear  in  mind  that  a  limit  is  a  general  limit,  unless  the  contrary 
is  stated.  Thus  in  2),  a;  runs  over  all  values  within  ^  as  it  =  a;  while  in  1)  it  range? 
only  over  a  specified  part  of  E. 


302  INDETERMINATE   FORMS 

2.  Let  f(x),  9(j^^  vanish  at  x=  a^  while  g(x)^^  within  RD(ji). 
Letf'{a)  exist,  finite  or  infinite.     Let  g' (a)  exist  and  be  =?^0.     Then 

For, 

/(a;)^        x-a        ^/(«) 

9 (P')     ^(^)  -  9i<^)      9'  (sO ' 
X—  a 

3.  We  can  generalize  2  as  follows : 

Let  f(x),  ^(^)  ^^^  their  first  n  —  2  derivatives  be  continuous  in 
B=  RI>(a}.  Within  B,  let  f^^-^^x)  be  finite  or  infinite,  g^''-'^\x~) 
finite,  and  g' ,  g"  •••  ^^"~^^9^ 0.  Let  /,  g  and  their  first  n—\  deriva- 
tives vanish  at  a.  Let  f^'^\a')  be  finite  or  infiriite,  while  g"^(a)  is 
finite  and  :#:0.      Then 

For,  by  449, 

f(x)^f^-'\c}^  c-a 

g(x)      g^--'\e)      g'^-'\c)  -  g'^-'^a^) ' 


But  as  x  =  a,  so  does  c  =  a.     Hence,  passing  to  the  limit  x  =  a, 
we  get  2). 

Example.     Let  f(x)  =  x^,  for  rational  x. 

=  0,  for  irrational  x. 

Let  g(x)  =  sin  x. 

Here  f'(x)  does  not  exist  except  at  x  =  0,  where  it  =  0.     Hence,  by  2, 

j;„„&L)  =  r(0)^0 

^=0  g{x)      g'(0)      1 

a  result  which  is  obvious  from  other  considerations. 

f(x') 
4.  In  1,  we  assume  the  existence  oi  X  =  R  lim     ,^  :^,  and  then 

show  that  ^  ^"^^ 

721ini4^=\.  (1 


THE   FORM  0/0  303 

That  the  limit  on  the  left  side  of  1)  can  exist  when  \  does 
not  is  shown  by  the  example  in  3.  It  is  also  illustrated  by  the 
following : 

Let  f(x)  =  x2  sin  - ,  for  x^  0. 


=  0,  for  X  =  0. 


Let  g(x)  =  X. 

Then,  f or  x  ^  0, 


while 
Hence 


f'(x)  =  2  X  sin cos  -: 

X  X 

g'(x)  =  1. 


x=o  g'(x) 
does  not  exist.    On  the  other  hand, 

lim4^  =  0. 

1=0  g{x) 

We  observe  that  this  result  also  follows  from  2. 

451.    Suppose: 

1°.  /(a;),  g(^x)  are  continuous  in  -D(+  oo); 

2°.  f  (x)  is  finite  or  infinite  in  D ; 

3°.  g'  (x)  is  finite  and  ^0  in  D; 

4°.  /(+oo)=^(+oo)  =  0. 

Let  lim     ,^      =  \,         X  finite  or  infinite. 

x=+=o  g  {x) 


where  x  runs  over  only  those  values  for  which  f  (jc)  is  finite. 


Then''  lim  4^  =  A-. 


We  set  1 

X  =  —' 

u 


Then  D  goes  over  into  R  =  BD*(0'). 

*  Cf.  footnote,  page  301. 


304  LNDETERMINATE  FORMS 

Let 


The  functions  <^,  i/r  not  being  defined  for  w  =  0,  we  set 

(/,(0)=t(0)  =  0.  (1 

Since  /,  g  are  continuous  in  i>,  ^,  t/t  are  continuous  in  i2,  by 
virtue  of  1)  and  4*^. 

For  points  of  I)  at  which  /'  (a;)  is  finite, 

^  ^  ->     dx  du         -^  ^  ^ 

Hence  at  the  corresponding  points  u  in  JR,  <^'  (u)  is  finite. 

■  From  the  relation 

A(^  _  A/    Aa; 

Alt      Aa;    Am' 

we  see  that  when  f  (x)  is  definitely  infinite  in  Z),  ^'  (u)  is  also 
infinite  at  the  corresponding  u  point  in  R. 

Thus  ^'(w)  is  finite  or  infinite  in  B,  while  -y^r' (u)  is  finite  and 
^  0  there. 

Then  by  450,  1,  if 

R  lim    , ,,  {  =  X,         \  finite  or  infinite. 
«=o   ->/^  (w)  -^  -^ 

u  running  over  only  those  points  for  which  <^' (u)  is  finite, 


i21im^=X. 


«=o 


^W 


But 

Also 


R  lim  7^  <^  =  lim  —~4--  (1 

„=o   '»/r('M)      ^=+«>  g{x) 


X  =  i2  Inn  ^f^  =  hm    :\)  {  =  hm  ^^yf^.  (2 

Hence  1),  2)  give  the  theorem. 


THE   FORM  c»/oo  305 

The  Form  g- 

452.    Let  f{a  +  0),  g{a  +  0)  he  infinite. 
In  R  =  RD{a)  suppose  that 

1°.  /(a;),  g{x^  are  continuous; 
2°.  f  (x)  is  finite  or  infinite  ; 
3°.  g' (x)  is  finite  and  =^0. 

f'(x) 
R  lim     ■  ;  ,  =  \,         X  ^mVe  or  infinite. 
.=a  g'ix) 

X  ranging  over  only  those  values  for  which  f  (x)  is  finite. 

Then  * 

R  ii,n  44  =  ^• 
x=a  g{x) 

Let  a<x<b<a-\-S.     Then,  by  448,  — f — i    i    i     .'^^    J^g  • 

(/(a:)      K-f)      /(I)  I         K^)i 


Thus 
whence 


Here  J  is  any  fixed  point  in  R. 

There  are  two  cases  according  as  \  is  finite  or  infinite. 
Suppose  \  is  finite.     Let  o-  >  0  be  small  at  pleasure  ;  we  can  take 
B  so  small  that  „,   ^^ 


y(D 


=  \  +  o-'.  \a'\<a: 


Let  T  >  0  be  small  at  pleasure.     We  can  choose  a  +  ?;  <  5,  such 

gix) '  gCx) 

are  numerically  <t,  ifx<fl!4-^<J- 

*  Cf .  footnote,  page  301. 


306  INDETERMINATE   FORMS 

Then  for  all  x  in  (a*,  a  +  ?;),  we  have  by  1), 


=  t'  +  (\  +  o-')(1-t").  (2 


Thus 

""'  -\|<T(l+|X|)  +  o-(l  +  T)<e, 


if  a-  and  t  are  taken  sufficiently  small. 

Suppose   \   is   infinite,  say  A,  =  +  oo .     Let   My^  0   be   large   at 
pleasure.     We  can  choose  S  so  small  that 

^  =  iJf(l  +  )^),         /.>0. 
Choosing  r]  as  before,  we  have  for  every  x  in  (a*,  a  +  t;), 

^  =  r'  +  if(l  +  /.)(l-T"). 
If  we  suppose  t<1,  and  TJf  sufficiently  large, 

^>iJf(l-T)-l>a,        inD/, 
where  Gr  is  as  large  as  we  please. 

453.    Xei/(+Qo),  ^(+Go)  he  infinite. 
In  D(+oo},  let 

•  1°.  f{x~),  g{x)  he  continuous; 

2°.  /'(a;)  he  finite  or  infinite; 

3°.   ^(a;)  he  finite  and  4zO. 

lim  -^  )  -^  =  \  X  iinite  or  infinite. 


!Z%gw 


We  deduce  this  theorem  from  452  in  the  same  way  as  451  was 
derived  from  450. 


THE    FORMS   0  •  oo,  oo-oo,  Qo,  1",  oo"  307 

Jhe  Forms  0  •  oo,  oo-oo,  0",  1",  oo" 

454.    1.   Let/(a;)=0,   g(x)=±cx^.     Then  f(x)g(x)  is   of   the 
form  0  •  GO. 

Setting  /^  =  Y»  tliis  form  is  reduced  to  ^• 

2.  Let  f{x)  =  ±  GO,  g(x)  =  ±  go,  the  infinities  having  same  signs. 
Then  f{x)  —  g(x)  is  of  the  form  oo  —  oo. 
Setting 


f-9  = 


this  form  is  reduced  to  — 

0 


1_1 

9      1 

fg 


3.  Let  f=0,  g=  0.     Then  [/(2;)]^(^)  is  of  the  form  O^. 

Let  y=/^        /(x)>0. 

Then 

log^  =  ^log/=-f- 


is  of  the  form  -• 

If  \ogy  =  \ 

then  lim  y  =  lim  [/(a;)]^^^^  =  e\ 

The  other  forms  1",  oo^  are  treated  in  a  similar  manner. 

EXAMPLES 
1.  a;'*log(l  —  cosx),        fx,  x^O.  (1 

has  the  form  0  •  oo  for  x  =  0.     We  may  write  it 

log(l  —  cos  x) 

which  has  tne  form  ^.     The  conditions  of  452  being  satisfied,  we  differentiate 
numerator  and  denominator,  getting  as  new  quotient 

sinx 


1  — cosx_     1  x^+^ 
2 


308  INDETERMINATE   FORMS 

This  has  the  form  -,  for  x  =  0.     Applying  450,  1,  we  get,  differentiating  once 
more. 


-2 


fX.+  I      xf^ 


^     sec2* 
2 
whose  limit  for  x  =  0,  is  0. 

Hence, 

i?  lim  x*^  log(l  —  cos  x)  =  0. 

1=0 

2.  /ilimx«|logx|'^  =  0.         a, /x>0.  (2 

x=0 

I'or,  ,,        ,„      llogxl'"      f(x) 

'  x''logx>"  =  '    °    '    =~^- 

'    °    '  x-«         g{x) 

We  apply  452. 

f'{x)  ^  IX  I  log  X  1^-1 
^'(x)      a       x-« 

If  /u  <  1,  this  expression  =0.     If  /x  >  1,  we  differentiate  again,  etc. 

3.  At  first  sight  one  might  think  that 

lim  /(^  +  ^)  -  1  n 

since  — ^^^-  =  1.     This  is,  however,  not  true  in  general. 
n 

For  example,  let  /(x)  =  e'. 

Then 

fin  +  1)  _  e"+i  _ 

/(n)      -   e"   ~^' 

Hence  the  limit  3)  is  here  e  and  not  1. 
Again,  let 


Then 

I 


/C«+i)_e^  _,,,_,, 


which  =  +  00. 

Criticisms 

455.    1.  The  treatment  of  indeterminate  forms  in  many  text- 
books is  deplorable. 

We  consider  some  of  the  objectionable  points  in  detail. 
When  /(a;),  g(x)  vanish  at  a;=  a,  the  function 

g(x) 
is  not  defined  at  a. 


CRITICISMS  309 

Some  authors  admit  division  by  0. 

From  this  standpoint  the  value  of  ^  at  a  is  hidden  because  ^ 

takes  on  the  indeterminate  form   —      The   true  value,  as    such 

authors  say,  may  often  be  found  by  a  simple  transformation,  or 
by  the  method  of  limits. 
For  example,  if 

f(x)=  x^  —  a\   g(x)  =zx—a, 

the  true  value  of  (f)  may  be  found  by  removing  the  common  factor 

a;—  a  in 

x"^  —  a^      ,  ^  X  —  a 


X—  a  ^  X  —  a 

Thus 


(x  +  a^ 


As  already  remarked,  division  by  0  is  ruled  out  in  modern 
analysis. 

First,  because  it  is  nowhere  necessary ;  and  secondly,  because 
of  the  difficulties  and  ambiguities  it  gives  rise  to. 

The  expression  1)  has  then  no  value  assigned  to  it  for  x  =  a. 
We  may  therefore,  if  we  choose,  agree  that  in  all  such  cases  (\> 
shall  have  the  value 


lim 


9i^) 


when  this  is  finite.  Some  authors  do  this;  in  this  case  ^  has  a 
true  value  at  a.  However,  we  shall  make  no  such  convention  in 
this  work. 

2.   In  this  connection  let  us.  give  an  example  of  the  so-called 
paradoxes  which  arise  from  division  by  0. 
Let  x=l;  then  ™2      i  _         i 

Dividing  both  sides  by  a:  —  1,  we  get 

a;  -f  1  =  1, 
which  gives,  since  x=l,  o  _  i 

It  is  easy  to  see  where  the  trouble  arises. 


310  INDETERMINATE   FORMS 

When  e^O,  we  can  always  conclude  from 

ac=bc  (2 

tnatj  7  yet 

a  =  o.  (d 

If,  however,  <?  =  0,  we  cannot  always  conclude  3)  from  2). 
In  fact,  take  a=^b,  ii  c=0,  we  still  have 

ac  =  bo. 

456.    1.  To  find  lim  ^(x),  some  writers  proceed  thus.     From 

/(a  +  70-/(a) 

h 

they  conclude  that  -p/  /-  \ 

lim<^  =  '-^.  (1 


This  is  correct  if  /'(a),  ^'(«)  exist,  and  the  latter  is  ^0. 
If  both  are  0,  they  say 


(2 


is  still  indeterminate.     Applying  the  preceding  reasoning  to  2),  it 
follows  that  its  true  value  is  that  of 

This  last  step  would  be  permissible,  provided  the  first  step 
showed  that 

limrf)=lim  ^^^-  (3 

But  it  does  not ;  it  shows  only  that  1)  is  true,  and  even  here 
we  must  assume  that  g' {a^-=^0. 

In  order  to  take  this  second  step  correctly,  we  have  proved 
Cauchv's  theorem,  448.     Cf.  449,  450. 


CRITICISMS  311 

2.  In  this  connection  we  note  that  we  cannot  always  say  that 

=^  and  ^4f^ 

have  the  same  limit  for  a;  =  a,  when  /(a)  =  gift)  =  0. 

For  example,  let 

1 
f(x)  =  x^  sin  -,  g^x)  =  x.         a  —  0. 

Then  .,  . 

Km  4^  =  0, 

"^^'^^  fr^:^  1        1 

•^  ■ ;  ,  =  2 a; sin cos-         x>0. 

g' {x)  X  X 

has  no  limit  for  x=  0.     Cf .  450. 

457.  Some  writers,  using  the  relation  of  Cauchy, 

conclude  now  that 

limc^  =  ^> 

This  is  true  if  /'(x),  g'  (x)  are  continuous  at  a,  and  ^  (a)=^0. 

458.  Some  writers,  in  order  to  evaluate  lim  0,  develop  /(a:), 
g(pc)  into  infinite  power  series.  The  possibility  of  such  a  develop- 
ment is  established  only  for  a  few  simple  cases  in  many  text-books. 
For  example,  such  books  do  not  show  that 

sec  a:,     tana;,     e*'°^ 

can  be  developed  into  power  series ;  yet  they  give  examples  of 
indeterminate  forms  involving  these  functions. 

There  is,  however,  no  necessity  of  using  infinite  series ;  all  that 
is  needed  for  such  cases  is  Taylor's  development  in  finite  form. 
See  445,  446. 


312  INDETERMINATE   FORMS 

459.    To  evaluate  the  form  ^,  some  writers  proceed  thus 

1 


Hence 

lim  c^C:.)  =  lim  ^  =  lim  <i>Kx^j^^ 

Dividing  by  lim  0(x),  they  get 

Hence  f'r\ 

g'{x) 

This  method  assumes  the  existence  of   lim  (^{x) ;    that  is,  the 
existence  of  the  very  thing  we  are  seeking  is  put  in  question. 
Suppose  by  this  method  we  find  that 

for  example;  what  right  have  we  to  say  that  therefore 

lim  4^  =  1? 

None  whatever,  until   by  some   subsidiary  investigation,   the 
existence  of  lim  ^  is  established.     See  377,  3. 

Scale  of  Infinitesimals  and  Infinities 

460.   Consider  the  functions 

f(x)  =  log"  X,    g(x)  =  x^,         a,  /S  >  0. 

Both  increase  indefinitely  as  x  =  +  <x>.     We  may  ask  which 
increases  faster. 


SCALE   OF   INFINITESIMALS  AND  INFINITIES  313 

The  quotient 

is  of  the  form  ^.       The   conditions  of   453   being   satisfied,  we 
consider 

^'      g\x~)      ^      x^     ' 

IfO<a<l,    ^1=0;  hence  ^=0. 
If  <x  >  1,  we  consider 

^2      g'\x)  /32  x^ 

Thusif0<«<2,   ^2  =  0;  hence  ^  =  0. 

If  a>2,  we  may  continue  this  process.  As  the  exponent  a  is 
diminished  by  unity  each  time,  log  x  must  have  finally  a  negative 
or  zero  exponent.     Thus  in  every  case  ^  =  0. 

461.  1.  Let/(a;),  ^(a;)  become  infinite  for  a;  =  a,  a  being  finite 
or  infinite.     If 

lim  '^^  ^  is  finite  and  =^  0, 
we  say  ^  and  g  are  of  the  same  order  infinite.     If 

we  say  y  is  of  lower  order  infinite  than  g.     If 
lim ''      '  is  infinite, 

we  say  /  is  of  higher  order  infinite  than  g. 

These  three  cases  are  denoted  respectively  by 

/(^)~^(a^).    /(«)<5'(2^).    /(a^)>5'(^)- 

We  may  also  say  more  briefly,  that /(a;)  is  infinitarily  equals  leSB 
than,  greater  than  gix). 


314  INDETERMINATE   FORMS 

2.  Similar  definitions  hold  when 

/(:r)  =  0,     ^(2;)  =  0. 
If,  for  example,  /.^  n 

we  say /(a:)  is  infinitely  small  relative  to  g{x),  or  an  infinitesimal 
of  higher  order  than  g(x).  We  may  also  say  f(x)  is  infinitarily 
smaller  than  gipc).     In  symbols, 

f{x)<g(ix). 

3.  Turning  to  the  result  of  460,  we  have : 

log"  x<x^       a,  /3  >  0,  2;  =  +  00. 
however  large  a,  is  and  however  small  /3. 

462.  Let  us  consider  now  functions  of  the  type 

For  a:  >  1,  we  have 

0  <  log  x<ix. 
For  sufficiently  large  x, 

l'1«//^  VniL'^  toti/      *  *  •      Vmtt/ 

are  >  0.     We  have,  then, 

The  values  of  these  iterated  logarithms  decrease  very  rapidly. 

For  example,  let 

a;  =1,000,000,000  =109. 
Then 

?ia:  =  20.723,  ^22^=  3.031,  ?3a;=  1.108,  ^42;=  0.103  ••. 

l^x  =  a  negative  number. 

Hence  l^x  does  not  exist. 

463.  1.    When  x=  +  00,  we  have,  if  a,  a^,  "2  "*■  ^^» 

re*  >  l-^^^x  >  l^'^x  >  l^'^^x  •••  (S 

The  sequence  S  may  be  called  the  logarithmic  scale. 


SCALE   OF   INFINITESIMALS   AND   INFINITIES  316 

To  prove  S,  let 

Then  w  =  +  oo  with  x.     We  have 

lim  — ^^^^ =  hm  _& =  0,         771  =  1,  2,  ••. 

by  461,  3. 

2.   /f  ofj,  a^,  •••  >1,  a:  =+00  ; 

l-f^x  >  l-^xl^^^x  >  l^xl^xl^^x  >  ••• 
This  follows  at  once  from  1. 

464.    Let  a;  =  +  00,  while  a,  «j,  a2,  •••  >  0.      Then 

a;"<  (e^)"i<  (g''"^)»2<  (e«''')"3< ...  (2» 

The  sequence  T  may  be  called  the  exponential  scale. 
Let  ^i^2-\ 

Ax~)  =  x%   g{x)=e'^^\    ^     J{JS 

We  apply  453.  f(^^«^_ 

g^  {pc)      ttje"!^ 
If  now  0<ct<l,  ^=0. 

If«>l,  /^(a:)_«(ft-l>°-2 

g"{x^~        ttjVi^ 

If  1  <  a<  2,  this  shows  that  ^  =  0,  and  so  on. 
Hence  ,    , 

To  show  ^   ^N  ^  ,=a:x„  ,- 

let  us  set 

u  =  e^. 

Then  ..      (g^Vi      „        M«h 

lim       /     =  lim  ^  „^    =  0, 

as  just  shown.     This  proves  1).     The  rest  of  the  theorem  follows 
now  in  the  same  way. 


316  INDETERMINATE   FORMS 

Order  of  Infinitesimals  and  Infinities 

465.  1.  Let  a:  =  0;  then  x  is  an  infinitesimal  j  x^^  t?  ••■  are  also 
infinitesimals. 

Taking  a;  as  a  standard,  we  may  say  x^  is  an  infinitesimal  of 
order  w,  n  being  a  positive  integer,  and,  in  general,  if  x>Q,  x*^  is 
an  infinitesimal  of  order  ytt,  where  ^l  is  any  positive  number. 

Then,  if 

i^lim^ 

x=0  X'^ 

is  finite  and  z^  0,  we  say  that  fix)  is  an  infinitesimal  of  order  fi. 
Not  every  infinitesimal,  however,  has  an  order. 
For  example,  by  464,  there  is  no  number  /u,,  such  that 

B  lim  — 

x=0      X'^ 
1 

is  not  0.     Hence  e  '  has  no  order. 

2.  On  the  other  hand,  an  infinitesimal  f^x)  may  not  have  an 
order  /*,  because 

x=0  X'^ 

either  does  not  exist,  or  when  it  does  it  is  infinite  or  zero. 
Thus 

X  sin  - 
E  lim 

1=0         x'^ 

does  not  exist.     Hence  ^ 

X  sin  - 

X 

is  an  infinitesimal  without  an  order. 

3.  Obviously,  similar  remarks  hold  for  infinities. 


466. 

of  21. 


CHAPTER   XI 

MAXIMA  AND  MINIMA 

ONE    VARIABLE 

Definition.     Geometric  Orientation 
Let /(a;)  be  defined  in  21  =  (a,  5).     Let  c  be  an  inner  point 


/  has  a  minimum  at  c.     If 

A/  =  /(:r)-/(c)<0ini)*(O, 


(2 


/  has  a  maximum  at  <?. 

In  words,  we  may  say :  /  has  a  maximum  at  c  when  /(c)  is 
greater  than  any  other  value  of  /  in 
the  domain  of  c;  it  has  a  minimum 
at  c  when  /(c)  is  less  than  any  other 
value  of  /  in  the  domain  of  e.  Accord- 
ing to  this  definition,  /(a;),  whose  graph 
is  given  in  the  figure,  has  a  maximum 
at  Cgi  ^-iid  a  minimum  at  Cj,  Cy 

The  reader  should  not  confuse  the  terms  f(x)  has  a  maximum 
or  a  minimum  at  a  point  c,  with  the  terms 

Min/(a;),  Max/(2;),  in  21. 

A  function  may  nave  an  infinite  number  of  extremes.,  that  is, 
maxima  or  minima,  in  2t. 


Example. 


f{x)  =  x-^{^ 


1  +  sin2  1^ , 

X 


=  0. 


for  X  9?:  0, 
for  X  =  0. 


317 


318 


MAXIMA    AND   MINIMA 


This  function  oscillates  between  the  two  parabolas 

y  =  x^,        y  —  2  x2. 

At  the  origin,  /has  a  minimum  ;  and  in  any  vicinity  of  the  origin,  /  has  an  in- 
finite number  of  maxima  and  minima. 


467.  We  consider  now  how  the  points  at  which  f(x)  has  an 
extreme  may  .be  determined.  Consulting  the  Fig.  in  466,  the 
reader  will  observe  that  at  the  points  of  extreme  the  tangent  is 
parallel  to  the  axis  of  x;  that  is,  at  these  points /'(a;)  =  0. 

However,  f(x)  does  not  need  to  have  an  extreme  at  all  the 
points  at  which /'(x)  =  0. 

For  example. 

This  is  an  increasing  function   whose   derivative  vanishes  at 
x=  0.     In  fact,  at  this  point  the  graph  has  a 
point  of   inflection   with  a  tangent  parallel  to 
the  X  axis.      See  Fig,   1. 

On  the  other  hand,  not  all  the  points  of  ex- 
treme are  given  by  the  roots  of  f  (x)  =  0. 

Example. 

y  =  x^. 

This  function  has  a  minimum  at  the  origin  0, 
which  is  a  cuspidal  point,  with  vertical  tangent. 
See  Fig.  2. 

At  this  point,  y  has  no  differential  coefficient, 
since 

■^/'(O)  =  +  00,      i/'(0)  =  -  oo.  Fig.  2. 


Fig.  1. 


Criteria  for  an  Extreme 
468.    1.  In  D(a'),  let  f'Xx^  he  continuous  andf^'''>(a)=^0.     Let 

Then  f  has  no  extreme  at  a,  if  n  is  odd.     If  n  is  even,  it  has  a 
minimum,  iff^"\a')>0;  a  maximum,  if  f"\a')<0. 


CRITERIA   FOR  AN   EXTREME  319 

For,  under  these  conditions,  we  have 

A/=/(a  +  A) -/(a)  =  J^ /(»)  (a  +  ^A). 

n ! 

Since  f^"\x)  is  continuous  at  a, 

sgn/(">(a  +  OK)  =  sgn/(»>(a)  =  a. 
If  w  is  odd, 

sgn  A/=  a  sgn  Ji. 

As  h  can  take  on  positive  and  negative  values,  A/  does  not  pre- 
serve one  sign  in  D*(a).  Hence,  /  has  no  extreme  at  a.  If  n  is 
even, 

sgn  A/=  0-. 
Thus  in  D*, 

A/>0,  if/«(a)>0, 

<0,  if /(«>(«)<  0. 

2.  iw  2t  =  (a,  5),  let  f  (x)  exists  finite  or  infinite.  The  points 
within  21  at  which  f(x}   has  an  extreme,  lie  among  the  zeros  of 

For,  suppose  /(.r)  has  a  maximum  at  c.     Then  for  A  >  0, 

/(c  +  70 -/C^X  0, /(c  -  70 -/(c)  <  0. 
Thus, 

/l  —  A 

But  when  7i  =  0,  • 

limi2  =  limX=/'(c).  (2 

On  the  other  hand,  1)  shows  that 

lim  -B  ^  0,  lim  i  ^  0  ; 

which  with  2)  shows  that  /'(c)=  0. 

3.  The  reasoning  in  2  also  shows : 

If  f{x)  has  an  extreme  at  x=  a,  thpnf'(a)=  0,  if  f  (cb)  exists 


320  MAXIMA   AND  MINIMA 

469.  Let  f(x)  he  continuous  in  D(a).  Let  f'(x)  he  finite  in 
D*(a~).     Let 

Ef'(a^  =  acc,Lf'{a')=  —  (rcc,   a  =  ±l. 
Then  f  has  a  minimum  at  «,  if  cr  =  +  1 ;   and  a  maximum,  if 

(T=-l. 

To  fix  the  ideas,  let  tr  =  +  1- 

ft=0  —  fl 

Thus  there  exists  a  S  >  0,  such  that 

/(a  +  A)-/(a)>0,         \h\<B, 
in  2>5*(a).     Hence  /  lias  a  minimum  at  a. 

470.  Let'f(x)  he  continuous  in  I)(cl).  In  D*(a),  let  f  (x)  he 
finite  or  infinite,  and  never  vanish  throughout  any  interval  of  it.  In 
RD*(ci),  letf'(x)  he  positive  when  not  zero  ;  in  LB* (a'),  let  f'(x) 
he  negative  when  not  zero.  Then  f{x)  has  a  minimum  at  a.  If 
these  signs  are  reversed,  f  has  a  maximum. 

For,  using  403,  we  see  that  f(x)  is  an  increasing  function  in 
RD(a),  and  a  decreasing  function  in  LD{a).  Hence  /  has  a 
minimum  at  a. 

EXAMPLES 

471.  -  fix)  = «  +  xi 

In  388,  we  saw 

i?/'(0)=+oo,        Z/(0)  =  -oo. 

Hence,  by  469,  /  has  a  minimum  at  0,  a  result  which  may  be  seen  directly. 

1 

472.  /(x)=^^,  forx:5fcO, 

=  0,  for  X  =  0. 

This  function  was  considered  in  366. 

Applying  470,  we  see  that  /  has  a  minimum  at  the  origin,  a  result  that  may  be 
seen  directly. 


CRITICISM  321 

473.  Let  /(x)  =  e  '\  for  xi=Q, 

=  0,       for  x  =  0. 

This  function  is  Cauchy's  function.  We  see  directly  that  it  has  a  minimum  at 
the  origin.    The  same  result  is  obtained  by  470. 

Criticism 

474.  Some  writers  confound  the  terms  the  function  has  a 
maximum  or  minimum  at  a  point,  with  the  terms  maximum  or 
minimum  of  a  function  in  an  interval.  These  two  terms  may  or 
may  not  mean  the  same  thing. 

For  example,  let 

f(x)=smx,         31[=(0,  2  7r). 

q 

Then  /  has  a  maximum  at  -,  and  a  minimum  at  -^.     These 

are  also  the  maximum  and  minimum  of/  in  21.     On  the  other  hand, 

if  we  take  ^  =  (0,  —  ]  as  one  interval,  /  has  neither  a  maximum 

nor  a  minimum  at  any  point  in  Sd  ;  yet  its  maximum  in  ^  is  1,  and 
its  minimum  in  ^  is  0. 

475.  The  following  example  also  illustrates  this  point. 

Find  the  greatest  and  least  distance  8  between  a  fixed  point  A 
within  a  circle  and  any  point  P  on  the  circle. 
Let  the  circle  be 

and  the  coordinates  of^lbea,  0;  a>0. 
Then 


h  =  -yj {X  —  cCf-  +  y^  =  Vrt^  -{-r^  —"l  ax, 

and  ,-, 

do —  a .  Q 

dx      ^ci^  ^r^-2ax 

Hence  S  is  a  decreasing  function  in  2l  =  (— r,  r),  and  has  no 
maximum  or  minimum  at  any  point  in  51.  It  has,  however,  a 
maximum  and  a  minimum  in  21,  viz. 


322  MAXIMA  AND  MINIMA 

SEVERAL  VARIABLES 

Definite  and  Indefinite  Forms 

476.  1.  Let  f(x-^  ••■  x„,)  be  defined  over  a  region,  of  which  a  is 
a  point. 

^^  A/  =  /(:r)-/(a)>0,         in  D*Ca% 

f  has  a  minimum  at  a.     If 

A/<0,         ini>*(a), 

/  has  a  maxim.um  at  a. 

The  theorem  of  468  may  be  generalized  thus : 

Let  the  partial  derivatives  of  f(x-^  ■  ■  •  x^i)  of  order  n-\-l  he  con- 
tinuous in  I>(ji).  Let  the  partial  derivatives  of  order  <n  vanish  at 
a,  while  the  derivatives  of  order  n  do  not  all  vanish  at  a.  Then  if 
71  is  odd,  f  has  no  extreme  at  a. 

Let  n  he  even.  If  d"f{a~)>  0  in  I)*(a^,  f  has  a  minimum  ;  if  it 
is  <  0,  it  has  a  maximum  at  a.  If  d"f(a)  has  both  signs  in  D*(jx), 
f  has  no  extreme  at  a. 

Let  iCj  =  a^  +  A  J . . .  a;^  =  a^  +  A^.     Then 

A/  =  -1-  d-fQa-)  +  _^  d-^'fia  +  eh-), 
n\  n  +  W 

by  434. 

Let  7;j  •••  t;^  be  the  direction  cosines  [244,  4]  of  the  line  L  join- 
ing a  and  x.     Then 

h-^  =  rT}^---hjn  =  rr]^, 

where 


Then  A/  =  f'H(ri^  +  r'^+^Kirj), 

since  d^f  d"'^]f  are  homogeneous  in  A^  •••  A„j. 

Since  the  derivatives  of  order  n-\-l  are  continuous  in  D^(a'), 
there  exists  a  positive  number  (r,  such  that 

\K\<a,         in  i>5(a).  (1 


DEFINITE   AND  INDEFINITE   FORMS  .        323 

If  now  (^y(a)>  0  in  D^Qa),  ^  is  >  0  on  the  sphere 

to  which  7]  is  restricted. 

Then,  by  355,  2,  there  exists  a  X  >  0,  such  that 

Then,  by  virtue  of  1),  we  can  choose  8'  <8  so  small  that 
A/>0,         ini),*(a). 

Hence  /  has  a  minimum  at  a. 

Similar  reasoning  shows  that  if  d^f(a)  <  0,  /  has  a  maximum. 

Consider  now  the  case  that  d"f(a}  has  both  signs  in  D*(^a). 
Suppose  that  it  is  positive  at  a  +  A  and  nega- 
tive at  a  +  k. 

Let  j&,  M  be  the  lines  joining  these  points 
with  a.     Let  r],  k  be  their  direction  cosines. 

^®*  R(iv)  =  A,         A>0. 

b:(k}=b,      b<o.    "" 

Let  a  +  th,  0  <  f  <  1,  be  a  point  on  L  between  a  and  a  +  h. 

Dist  (a,  a  +  h')  =  r;    then  Dist  (a,  a  +  th^  =  tr. 

^^^^  A/  =  /(a  +  th)  -  /(a)  =  t"r"  \  A  +  trK'  I . 

Let  tn  be  such  that  ^    ^       . 

^  t^rG  <  A. 

Then,  by  1),  A/>  0  for  all  points  on  L  between  a  and  a  -f-  i^A,  a 
excluded.  Similar  reasoning  shows  that  A/<  0  for  all  points  on 
iHf  sufficiently  neUr  a,  the  point  a  excluded. 

Thus  in  any  domain  of  a,  however  small.  A/  has  opposite  signs. 
Hence  in  this  case  /  has  no  extreme  at  a. 

Let  n  be  odd.     Then  IT  being  homogeneous, 

Hence  d^f^a)  has  opposite  signs  in  every  domain  of  a.  Hence 
when  n  is  odd,  f  has  no  extreme  at  a. 


324  MAXIMA   AND  MINIMA 

2.  Let  f(x^  •••  a:„j)  have  partial  derivatives  of  the  first  order ^  finite 
or  infinite^  in  the  region  R. 

The  points  of  i2,  at  which  f  has  an  extreme^  satisfy  the  system  of 
equatio7is  ,  „  .^ 

1^=0  ...  ^=0. 

The  demonstration  is  analogous  to  that  of  468,  2. 

477.  1.  We  have  just  seen  that  the  sign  of  c^"/(a)  plays  a 
decisive  r81e  in  questions  of  maxima  and  minima.  But  as  already 
observed,  d"f  is  a  homogeneous  integral  rational  function  of  h^ 
Agi  •••  h^  of  degree  n.  Such  functions  are  subjects  of  study  in 
algebra  and  the  theory  of  numbers,  where  they  are  often  called 
forms. 

A  form  ^(^x^  ••.  x^~)  which  has  always  one  sign,  except  at  the 
origin  where  it  necessarily  vanishes,  is  called  definite. 

Such  a  form  is 

a^%^+   ■■+ajxj,  (1 

the  a's  being  not  all  0. 

If  the  sign  of  a  definite  form  is  positive,  it  is  called  a  positive 
definite  form  ;  if  negative,  it  is  a  negative  definite  form.  Thus  1) 
is  a  positive  definite  form,  while 


4  •'^l  "'m  •'^m 


is  an  example  of  a  negative  definite  form. 

If  <I>  can  take  on  both  signs,  it  is  called  indefinite. 
Thus 

X^+-'-  +  xJ 

is  an  indefinite  form. 

There  is  a  class  of  forms  which  vanish  at  points  besides  the 
origin  and  yet,  when  not  0,  have  always  one  'sign.  They  are 
called  semidefinite  forms. 

Such  a  form  is,  for  example, 

{a^x^+ -  +  a^x^y, 
which  is  positive  when  not  0. 
Consider  the  quadratic  form 

F  =  Ax"^  +  2  Bxy  +  Cy^. 


DEFINITE   AND   INDEFINITE   FORMS  326 

If  A=^0,  we  can  write  it 

F=  i  j  (Ax  +  Byy  +  (AC  -  B2)y^ 

If  the  determinant 

D  =  AC-Bi 

is  >  0,  J?' does  not  vanish  except  at  the  origin,  and  is  therefore  a  positive  definite 
form  if  ^  >  0,  and  a  negative  definite  form  if  ^  <  0. 
If  Z>  <  0,  J?"  is  an  indefinite  form. 

F=^lAx  +  By\  . 

Hence  F  vanishes  on  the  line 

Ax  +  By  =  0, 

but  has  otherwise  one  sign.     Thus,  in  this  case,  F  is  semidefinite. 

2.  The  theorem  of  476  may  now  be  stated  as  follows  :  If  d"f(^a) 
is  an  indefinite  form,  f  has  no  extreme  at  a.  If  it  is  a  'positive  definite 
form^  f  has  a  minimum ;  if  it  is  a  negative  definite  form^  f  has  a 
maximum  at  a. 

478.  When  n  =  2,  i.e.  when  not  all  partial  derivatives  of  the 
second  order  are  0  at  a,  d'^f(cC)  becomes  a  quadratic  form., 

d^f(a')=  ^a^jiji^;         i,  /c=  1,  2,  •••  w.  (1 

where  j-t,      ^  n 

«c.=/     x,x,(«l     "O' 

and  hence 

The  determinant 


A    — 

^11    ^12    ■ 

•    «l7n 

^m  — 

^/nl    ^ni2    * 

^mm 

is  called  the  determinant  of  the  form  1). 

Let  Am_i  be  obtained  by  deleting  the  last  row  and  column  in  A^ ; 
let  A^_2  be  obtained  by  deleting  the  last  two  rows  and  columns  in 
A,  etc.;  finally,  let  Aq=  1. 

In  algebra  the  following  theorem  is  proved  : 

In  order  that  the  form 

.2/.^.^«  '  (2 

be  a  positive  definite  form,  it  is  necessary  and  sufficient  that  the 

signs  of  ...  „ 

^  Ao,   Ai,  ...  A,  (3 


326 


MAXIMA   AND   MINIMA 


are  all  positive.  For  2)  to  be  a  definite  negative  form,  it  is  neces- 
sary and  sufficient  that  the  signs  in  3)  are  alternately  positive  and 
negative. 

Applying  this  result  to  the  theorem  in  476,  we  have : 

Let  the  partial  derivatives  of  the  third  order  he  continuous  in 
D(a),  and  let  those  of  the  second  order  not  all  vanish  at  a.  Let  all 
the  first  derivatives  vanish  at  a.     Let 


r=0,  1, 
Ao=l. 


m. 


A.=    ! 

If  the  signs  in  the  sequence 

\.  Aj,  •••  A^ 

are  all  positive,  f  has  a  minimum.     If  the  signs  in  this  sequence  are 
alternately  positive  and  negative,  f  has  a  maximum  at  a. 


Semidefinite  Forms 

479.  Up  to  the  present,  the  case  that  d"f(a)  i-s  a  semidefinite 
form  has  not  been  treated.  It  is,  however,  easy  to  show  that  in 
this  case  /  may  or  may  not  have  an  extreme  at  a. 


Ex.  1. 


Here 


Hence, 


/(xy)  =  x^-Qxy'^  +  8yi  =  (x-2  y'^){x  -  4y2) 
a  =  (0,0). 

A(0)  =  0,    A(0)  =  0; 
/".<0)  =  2,    /%(0)  =  0,    /'V(0)  =  0. 
d2/(0)  =  h-^. 


We  have  here  a  semidefinite  form. 
That  /  has  not  an  extreme  at  the  origin  is 
obvious. 

For,  if  P  is  the  parabola 

a;  =  2  2/2, 
and  Q  the  parabola 

cc  =  4  2/2^ 


f>0 


we  see  that  /<  0  between  these  parabolas,  and 

>  0  in  the  rest  of  the  plane,  points  on  the  parabolas  excepted,  as  in  the  figure. 


CRITICISM  327 

Ex.2. 

f{xy)  =  2/2  +  x^y  +  x* 

x2\2  ,  3 


=  (..f)%f..  a 


a  =  (0,  0). 

Obviously,  from  1),  /  has  a  minimum  at  the  origin. 
Here 

/x'(0)  =  0,   /;(0)  =  o. 

/i;i(0)  =  0,    /i^(0)=0,    /;i(0)  =  2. 
Hence, 

d2/(0)  =  k\ 

We  have  here  a  semidefinite  form. 

480.  It  is  beyond  the  scope  of  this  work  to  do  more  than  show 
that  the  semidefinite  case  is  ambiguous  and  requires  further 
investigation.  We  refer  the  reader  for  a  detailed  treatment  of 
this  case  to  Stolz,  Grrundziige^  Vol.  1,  p,  211  seq. ;  Jordan,  Gour%.^ 
Vol.  1,  p.  380  seq.\  Scheeffer,  Math.  Ann.,  Vol.  35,  p.  541; 
V.  Dantscher,  Math.  Ann.,  Vol.  42,  p.  89. 


Criticism 

481.  The  partial  derivatives  of  order  n  being  continuous  in 
D(a),  we  have  seen  that 

A/=  ^/(a)  +  |j  (^y(a)  +  -  +  ^1;  <^"/(«  +  ^A)  =  ^1  +  n  +  -  +  T,- 

The  terms  T^,  T^,  •••  are  polynomials  in  Ap  h^,  •■•h^  of  1°,  2°,  ••• 
degree,  whose  coefficients,  except  the  last,  are  constant.  Letting 
Aj,  ^2,  •••  be  infinitesimals  of  the  1°  order,  ^^  if  ^0  is  thus  an  in- 
finitesimal of  rth  order.  The  assumption  is  now  made  by  many 
authors  that  if  r  <.s,  then  T^  is  infinitely/  small  compared  with  T^. 
When  there  is  only  one  variable  h,  this  is  indeed  true ;  it  is  not, 
however,   always   true   when   there   are   two   or  more    variables 

As  an  example,  consider  the  form 


328  MAXIMA   AND   MINIMA 

Let  the  increments  Aj,  h^  be  related  by  the  equation 

K^^Jh\  (1 

i.e.  let  the  point  (A^  h^)  approach  the  origin  along  the  parabola  1). 
Then 

which-shows  that  T^,  T^,  instead  of  being  infinitely  small  compared 
with  7^2'  ^^6  in  fact  numerically  6  and  8  times  larger  than  T^. 

482.  Let  us  see  now  how  this  erroneous  assumption  regarding 
infinitesimals,  when  applied  to  the  semidefinite  case,  leads  to  a 
false  result.*     For  simplicity  we  take  only  two  variables  x,  y. 


=  1  lAJfi  +  2Bhk  +  CB\  +... 


2! 

=  ^2+^3+- 

Let  the  determinant  AC  —  &=0.     Then  T^  is  a  semidefinite 
form.     To  fix  the  ideas  let  ^  =5^  0  ;  then 


2^1 


^2  =  A  l^A  +  Bk\^      by  477,  1. 


Thus  A/  has  the  sign  of  J.,  except  for  the  points  (A,  ^)  on  the 

line  L, 

Ax  +  By  =  0.  (i 

For  points  on  i,  A/  becomes 

For  points  on  the  line  i,  on  opposite  sides  of  the  origin,  T^  takes 
on  opposite  signs.  As  the  sign  of  A/  at  these  points  depends  on 
the  sign  of  T^  (making  use  of  the  above  erroneous  assumption), 
it  is  thus  necessary  that  jPg  =  0  for  points  on  i,  if  /  is  to  have 
an  extreme  at  a,  6.     If  ^g  =  0  for  these  points, 

*  Cf .  TocUiunter,  Differential  Calculus ;  Desmartes,  Cours  d' Analyse. 


RELATIVE   EXTREMES 


329 


for  these  points.     If  now  T^i^O  on  i, / has  an  extreme  *  at  (a,  6), 
if  T^  has  the  same  sign  as  A,  for  points  of  L,  the  origin  excluded. 
That  this  result  is  wrong  may  be  shown  by  applying  it  to 

which  we  considered  in  479,  Ex.  1. 

Here 

^2  =  7^2,  T^  =  -6hk^  T^  =  8k\        A  =  2. 

The  line  L  is,  in  this  case,  the  ?/-axis.  For  points  on  X,  T^  =  0; 
while  T^  >  0,  the  origin  excepted.  Thus  T^  has  the  same  sign  as 
A.  We  should  have  therefore  an  extreme  at  the  origin,  if  the 
above  reasoning  were  correct.  But  as  we  already  saw  in  479,  / 
has  no  extreme  at  the  origin. 

483.  Another  error  which  is  sometimes  made  is  the  following. 
It  is  assumed  that  the  function  /(a;,  y)  has  an  extreme  at  the  point 
R  when  and  only  when  /  has  an  extreme  along  every  right  line 
through  JR.. 

That  this  view  is  incorrect  is  seen  by  the  function 

/(^W)  =  (a:  -  2  /)(a;  -  4  /), 

given  in  479,  Ex.  1. 

As  the  figure  shows,  a  point  *S' 
moving  along  any  line  L  toward  the 
origin  72,  finally  remains  in  a  region 
for  which  A/>  0,  the  origin  of 
course  excluded.  If,  therefore,  this 
view  were  correct,  /  would  have  a 
minimum  at  M,  whi'ch  we  know  is 
not  true. 

Relative  Extremes 

484.  1,.  Let  us  consider  the  problem  of  finding  the  points  of 
maxima  and  minima  of  a  function  ^ 


*  According  to  the  above  erroneous  hypothesis. 


(1 


330  MAXIMA    AND   MINIMA 

where  the  variables  u^  •••  Up  are  one-valued  functions  of  x^  •••  a;^, 
defined  over  a  region  M  and  satisfying  the  system 

.         •         •         •  (2 

Such  points  of  maxima  and  minima  are  called  points  of  relative 
extreme,  to  distinguish  them  from  the  case  when  the  variables 
Xj^  ■••  x^,  u^  •••  Up  are  all  independent. 

Let  the  point  (a^j  •  •  •  Up)  run  over  the  region  T  when  Qx^  •  •  •  a;„) 
runs  over  M.  Let/,  ^j  •••  <f)p  have  continuous  first  partial  deriv- 
atives with  respect  to  a^j  •••  Up  in  T,  and  let  the  w's  have  continu- 
ous first  partial  derivatives  with  respect  to  a;j  •••  x^  in  M. 

Let  w  considered  as  a  function  of  a^j  •  •  •  x^  be  denoted  by 

w  =  F(x^  •••  ^m)- 
The  points  of  extreme  of  w  in  R  satisfy  the  system 

M!  =  0.-^=0,  (3 

dx^  dx^ 

by  476,  2.  Let  S  denote  the  set  of  points  determined  by  3). 
Lagrange  has  given  a  method  for  forming  the  system  3),  which 
is  often  serviceable.  It  rests  on  the  introduction  of  certain  unde- 
termined multipliers  /ij  •••  /x^.  In  fact,  differentiating  1)  2)  we 
get: 

du^  =  J-dx,^...^J-Jx^^J^du,^...^J-dUp, 


dx-^       ^  dx^  dUi       ^  dUp 

Multiply  the    2d,  3d,  •••  equations   by  /Xj,  fi^,  •••  respectively, 
and  add  the  results  to  the  first  equation. 
We  get 


RELATIVE   EXTREMES  331 

Let  us  now,  if  possible,  determine  the  /a's  so  that 

f  +  X^.p  =  0,-f+Xt^r^^  =  0.  (5 

Then  4)  gives 
Then,  by  429,  2, 


These  are  the  left  hand  members  of  the  equations  3). 
Hence,  by  3), 

dx^       -^    ''  dx^         '        dx^      ^    ''  bx^ 
Thus  the  points  of  S  are  determined  by  2),  5),  73. 
2.  Let  us  introduce  the  function 

^  =/+/*!</>! +  •••  + /*P<^p- 

We  observe  that,  considering  x^  ••■  Up  as  independent  variables, 

1^  =  0,  ...^  =  0,         1^  =  0,...  1^  =  0  (8 

dXj  ox^  dMj  dUp 

are  precisely  the  equations  5),  7). 

485.  To  determine  whether  a  point  of  *S'  is  a  point  of  extreme, 
it  is  often  necessary  to  consider  the  second  and  even  higher  differ- 
entials of  Fl^x^  •••  Xjn).  Here  it  is  sometimes  convenient  to  make 
use  of  the  fact  that 

where  the  differential  on  the  right  is  calculated  supposing 


h  '"  "^mi  ■**! 


to  be  independent  variables. 


332  MAXIMA   AND  MINIMA 

For,  let  us  denote  g  considered  as  a  composite  function  of  x  by 
G-i^x^  •••  x^^.     Then 

F{x^-- x^)=a(x^- x^\ 

since  the  <^'s  vanish  now  by  484,  2.  • 

Hence 

But,  by  433,  2), 

=  (^V.by484,  8). 

486.    Example.    Let  us  find  the  shortest  distance  from  the  point  P  =  {axa^az)^ 
to  the  plane 

<t>  =  biXx  +  62^:2  +  ^'3X3  +  60  =  0.  (1 

Let 

to  =  52  =  S(a;,  -  a,)2        i  =  l,  2,  3. 

L 

=  f{XiX2Xz). 

The  points  of  minimum  value  are  the  same  for  w  and  5. 
We  have 

</  =  /  +  M<^, 
^  =  2(x,  -  aj  +  M&.  =  0.         i  =  1,  2,  3.  (2 

From  the  four  equations  1),  2),  we  find 

Xi  —  Ot  =  -  I  fib^, 

_  n  ai&i  +  a-jb^  +  o-sba  +  bp 
'^~  bi^  +  62^  +  632 

Hence  at  this  point  x  =  ^, 

_  (ai&i  +  a2&2  +  mbs  +  60)^  ,q 

612  +  622  +  632  ■  ^ 

To  ascertain  if  ^  is  a  point  of  minimum,  consider  the  value  of  cPw  at  this  point. 

dg  =  'L  {2{x,  -  a,)  +  iib,}  dx,,        t  =  1,  2,  3. 

d2g  =  2I,  dx,2. 

As  this  is  positive,  |  is  a  point  of  minimum.     Thus  the  least  distance  5o  from  a 
to  <p  is  determined  by  3).     We  get 

.  _  aibi  +  a262  +  (isbs  +  60  • 

°~       V  6i2  +  632  +  632 


CHAPTER   XII 


INTEGRATION 

Geometric  Orientation 

487.  1.  As  the  reader  is  probably  aware,  the  integral  calculus 
arose  from  attempts  to  find  the  length  of  curves,  the  area  of  sur- 
faces, and  the  volume  of  solids. 

Before  taking  up  the  general  theory  of  integration,  let  us  see 
how  the  problem  of  finding  the  area  of  a  simple  figure  leads  to  an 
integral. 

2.  In  the  interval  21  =  (a,  5)  let  y  =  f{x)  be  an  increasing  con- 
tinuous function  whose  graph  F  is  given  in  the  adjoining  figure. 

We  seek  the  area  A  of  the  figure 
aba^  =  F.  The  upper  boundary  of 
F  is  the  curve  F. 

To  find  the  area  of  a  circle  in 
elementary  geometry,  we  form  a 
sequence  of  inscribed  and  circum- 
scribed polygons.  Each  inscribed 
polygon  is  contained  in  the  circle ; 
each  circumscribed  polygon  contains  the  circle.  We  say  the  area 
of  the  circle  is  therefore  less  than  any  of  the  outer  polygons,  and 
greater  than  any  of  the  inner  polygons.  We  then  show  that  the 
areas  of  these  two  systems  of  polygons  have  a  common  limit  as 
the  number  of  the  sides  increases  indefinitely.  This  limit  is  then, 
by  definition,  the  area  of  the  circle. 

We  shall  adopt  a  similar  procedure  here,  reserving  for  later  a 
more  thorough  discussion  in  connection  with  other  fundamental 
geometric  notions. 

Let  us  divide  (a,  5)  into  n  equal  intervals  by  introducing  tlio 
points  «!,  a^^  «3>  ••• 

333 


334  INTEGRATION 

Over  each  interval  (a^,  a^+i)  ^^^  have  two  rectangles 

and  „    _  ,, 

Let 

Then  /Sn  contains  F,  while  s„  is  contained  in  F, 
Let  us  now  divide  each  of  the  intervals 

(a,  ftj),   («i,  ^^a)'   •••(«n-i'^) 

,  into  two  equal  parts.  We  get  two  new  sums  S2n  and  iS'g^.  In  this 
way  we  may  continue  without  end.  Let  us  now  give  n  the  values 
1,  2,  4,  ... 

We  get  two  limited  univariant  sequences 

Each  sequence  has  a  limit  by  109.  These  limits  are,  moreover, 
the  same.  For  *S'„  —  s„  is  obviously  the  area  of  the  shaded  region 
in  the  figure. 

b  —  a 


Hence 

Evidently 
Hence 


'S'n  -  S„  =  (/3  -  «)       ^ 

lim(*S'„-O=0. 
lim  *S'„  =  lim  s„. 


As  in  the  case  of  the  circle,  the  common  limit  is,  by  definition, 
the  area  of  F. 

3.  Now  T      ^ 

0  —  a 


Hence  » 

0  —  a 


s„  = 


n 


rm=——f(^rn)' 


f/(a)+/(ai)+-+/(«^«-i)|;  0 


ANALYTICAL   DEFINITION   OF  AN   INTEGRAL  335 

and  therefore,  setting  for  uniformity,  a^  =  a, 

^  =  lim2/(0^-  (2 

But  as  the  reader  knows,  the  expression  on  the  right  of  2)  is 
the  integral  ^j 

I  f(x)dx. 

488.    Example.    Let  us  find  the  limit  A  of  487,  2)  for  the  function 

/(x)=c^.        c>0. 
6  —  a 


Set 


then 
and 
Hence 


n 
a„  =  a  +  m5,        m  =  0,  1,  •••  n  —  1. 

s„  =  5  .  c«{l  +  c«  +  c2«  +  ...  +  c("-i)«} 


Now 

Also,  by  311, 

From  1),  2)  we  have 


=  5.c«l^^  =  (c^-c»)-l—  (1 

1  _  c«      ^  ^  cs  -  1  ^ 

lim  5  =  0. 
lim— i— =  -i-.  (2 

5=0  C*  —  1       log  C 

^  =  lim  s„      <^  ~  ^ 


logc 


Analytical  Definition  of  an  Integral 
489.    1.   Let  f(x)  be  a  limited  function,  defined  over  the  inter- 


H — \- 


val  21  =  (a,  6),  a  <  6.  , ^ ^ ^ ^ ^ , 

a       ttj  ttj     Qj  a-n-i  0 

Let  us  divide  21  into  n  sub-intervals 

K  =  («m-l«m).  m  =  1,   2,    ...   W 

by  interpolating  at  pleasure  the  points 

For  uniformity  of  notation,  we  set 

a  =  a.Q,     h  =  a^. 


336  INTEGRATION 

This  set  of  points  1)  produces  a  division  of  51,  which  we 
denote  by 

Since  no  confusion  can  arise,  let  S,„  denote  also  the  length  of 
the  interval  S^.  The  greatest  of  these  lengths  we  call  the  norm 
of  D  and  denote  it  by  8. 

In  each  3^,  let  us  take  a  point  |^  at  pleasure,  and  build  the  sum 

^8  =  2;/(l™)(«^-«m-i)  =  2/(IJC  (2 

In  passing,  let  us  note  that  the  sums  S,^,  s„  of  487  are  special 
cases  of  2). 

Let  now  8=0.  If  ^T^  converges  to  a  limit  J,  which  is  independ- 
ent of  the  choice  of  the  points  a^,  |^,  we  write 

J=  lim  V/(^^)S^  =  ffCx^dx. 

We  say  f(^x)  is  integrahle  from  a  to  b,  and  call  J  the  integral  of 
fix)  from  a  to  h.  f(x)  is  called  the  integrand;  a,  h  are  respec- 
tively the  lower  and  upper  limits  of  integration.     We  also  write 

J=  ffdx. 
The  symbol  J   is  a  long  S^  the  first  letter  of  the  word  sum. 

2.  To  fix  the  ideas,  we  have  taken  a<b. 
Then  in  2),  the  numbers 

are  positive. 

If  we  had  taken  a>b,  the  3's  would  be  all  negative.  Evidently, 
whether  a  is  greater  or  less  than  b,  if 

ffCx^ldx  (3 

exists,  then  '^'' 


£f(^')d^  (4 


exists  and  3),  4)  have  the  same  numerical  value,  but  are  of  op- 
posite sign. 


UPPER   AND   LOWER  INTEGRALS  337 

Without  loss  of  generality,  we  may  therefore,  in  our  discussion, 
take  a<  J  so  that  the  S's  are  >  0. 

3.  Obviously  the  symbol 

has  no  sense  when  a  =  b.     In  this  case  we  shall  assign  to  it  the 
value  0. 

4.  Letf(x)  he  integrahle  in  %  =  (a,  6),  and 

Then 

I  Cf^xWlMih-a).  (5 

For, 

or, 

-  M(h  -a)<J^< M(h-  a). 

Passing  to  the  limit,  8=0, 

-  M(h  -a)<  Cfdx  <  M(h  -  a), 
which  is  5). 

Upper  and  Lower  Integrals 

490.    Before  deducing  criteria  for  the  integrability  of  /(a;),  we 
define  upper  and  lower  integrals. 

^^*  M^  =  Max/(a;),         m^  =  Min/(a:),  in  h^. 

We  shall  show  immediately  that  the  limits 
aS'  =  lim  Sj)^         S  =  lim  Sd 

5=0  6=0 

exist  and  are  finite.     They  are  called  respectively  the  upper  and 
lower  integrals  of /(a;)  for  the  interval  51,  and  are  denoted  by 


i  fdx,  i  fdx. 

J%  «/9I 


338  INTEGRATION 

491.  If  f(x)  is  limited  in  21,  the  upper  and  lower  integrals  exist 
and  are  finite. 

Let  us  consider  the  upper  integral;  similar  reasoning  is  appli- 
cable to  the  lower  integral. 

Corresponding  to  each  division  D  there  is  a  Sq.  Let  us  lay 
off  these  values  on  an  axis.      We  get  a  limited  point  aggregate. 

• 1 + 1 1 — 


For,  since  f(x)  is  limited,  there  exists  a  number  M>  0,  such 
that 

-MKfix^KM. 

or  -  M(h  -a}<S^<  M(h  -  a). 

Hence,  the  Sj)  are  limited. 

Let  Sq  =  Min  Sj^. 

We  show  now  that  _  _ 

Sq  =  lim  Sj) ; 

5=0 

that  is,  for  each  e  >  0,  there  exists  a  8q,  such  that 

S^-S,<e  (1 

for  any  division  D  of  norm  S<8q. 

Since  Sq  is  a  minimum,  there  exists  a  division  A  of  51  of  norm 
97,  such  that  _       _       _ 

>S'o<'S'^<'^o  +  f-  (2 

Let  7/ J,  772,  •••  rj^  be  the  intervals  of  A.     Let 

11'        u'        is' 

be  the  intervals  of  D  lying  wholly  in  77^,  t  =  1,  2,  •  •  •  v ;  let 

a;.  Si,  ... 

be  the  other  intervals  of  D. 


UPPER   AND   LOWER  INTEGRALS  339 

We  take  now  8^  so  small  that 


for  all  8  <  8q.     This  is  evidently  possible,  since  A  has  only  v  inter- 
vals, V  being  fixed. 

I.et 

JM:,=  Max/,  in  S,„ 


Then 


or. 


M[  =  Max/,  in  8[. 

<^^  +  |,  by3).  (4 

Hence,  from  2),  4), 

Sji-S^^<e,         8<8f^.  Q.E.D 


492.   Ex.  1. 


Then 


2t=(0,  1). 
fix)  =  1  for  rational  points  in  21, 
=  0  for  irrational  points  in  %. 
8d  =  1,         Sd  =  0. 
8-1,  8  =  0. 


340  INTEGRATION 

Ex.  2. 

Let  /(x)  lie  on  the  circumference  of  the  circle  for  rational 
X,   while  it  lies  on  the  edge  of  the  inscribed  square  for     ~^  0 

irrational  x. 

Then  evidently  _ 

4    fdx  =  ^  irB'^,  area  of  semicircle  ; 
J  91 


J   fdx  =  B'^,  area  of  half  the  inscribed  square. 
5)r 


Criteria  for  Integrability 

493.    For  the  limited  function  fix)  to  he  integrahle  in  the  interval 
5t,  it  is  necessary  and  sufficient  that 


//^^^//^' 


It  is  sufficient.     For,  let  D  be  any  division  of  norm  8. 

Then 

M^>faj>m^. 
Hence 

Summing,  we  get 

By  hypothesis, 

lim  Sj)  =  lim  Sj)  =  L,  say. 

S=0  6=0 

Hence 

lim  Jg  =  L. 

It  is  necessary.     For,  since  the  integral  J"  exists, 

e>0,     S,>0,     1-^-2/^)8  I  <|  (1 

for  any  division  D  of  norm  S  <  8^- 

Let  7}^  be  any  other  point  in  the  subinterval  B^  belonging  to  the 
division  D  just  mentioned. 

We  have  also,  as  in  1),  - 

(2 


|J--E/(OS.|<^. 


CRITERIA  FOR  INTEGRABILITY  341 

Subtracting  1),  2),  we  have 

1 2/(105.- 2/(7;  J8 J  <|,  (3 

for  any  division  of  norm  8  <  8q. 

On  the  other  hand,  in  each  S.  there  are  points  f^,  t/^,  such  that 

4(6  —  a) 

4(6  —  a) 
Multiplying  these  inequalities  by  8^  and  adding,  we  have 


Hence 


S,=  ^mA>^f(ivJK-l- 


S^  -  S,  <  S/(fOS,  -  ^fCvjB^  +  |- 


This  gives,  using  3), 

Hence 

S-S<€;    hence  S=S. 

494.  We  can  state  the  theorem  of  493  a  little  differently  by 
introducing  the  following  definitions.  The  difference  between 
the  maximum  and  minimum  of  a  function  /{x)  in  an  interval  31, 
is  called  the  oscillation  of  f  in  21.  It  cannot  ever  be  negative. 
Let  D  be  any  division  of  %  into  subintervals  S^,  of  length  8^.  Let 
(Ok  be  the  oscillation  of/  in  8^.     The  sum 

is  called  the  oscillatory  sum  of /for  the  division  D. 
We  have 

n^f=  2  (M^  -  mj  8^  =  ^MA  -  2mA        ' 

=  Sj)  —  Sp,  (1 


342  INTEGRATION 

495.  In  order  that  the  limited  function  f(x)  he  integrahle  in  21,  ii 
is  necessary  and  sufficient  that 

limn/=0.  (1 

For,  by  494,  1), 

Qf^iS/)  —  Sj). 

By  493,  /(a:)  is  integrable  when  and  only  when 
lim  xS'^,  =  lim  >S'^, 
or  when  and  only  when 

5=0 

which  is  1). 

496.  if  /(a;)  is  integrable  in  51,  it  is  integrable  in  any  partial 
interval  Sd  of^. 

Let'  2l  =  (a,  5),  «  =  («,  yS).     c      I  I         6 

Since 

lim  2a)^S^  =  0  (1 

6=0     D 

for  any  system  of  divisions  whose  norm  8=0,  let  us  consider  only 
such  divisions  involving  the  points  a,  /3.  Let  D^  be  the  division 
of  ^,  produced  by  D.     Then 

Z)j  D 

since  the  first  sum  contains  only  a  part  of  the  intervals  S^,  and  ©^  is 
Passing  to  the  limit  in  2),  we  have,  by  1), 


lim  2&)^8^  =  0. 

S=0     D 


Hence, /(a;)  is  integrable  in  ^. 

497.  In  order  that  the  limited  function  f(pc)  be  integrable  in  51,  it 
is  necessary  and  sufficient  that,  for  each  e  >  0,  there  exists  at  least  one 
division  D  for  which  _    „ 

That  this  condition  is  necessary  follows  at  once  from  495. 


CRITERIA   FOR  INTEGRABILITY  343 

It  is  sufficient.     For,  by  494,  for  the  division  D 

' "s ^J ' 

But  then,  as  the  figure  shows,  ~**  ° 

or,  by  491, 

But  then,  by  87,  5, 

Therefore,  by  493,  f(x)  is  integrable. 

498.  In  order  that  the  limited  function  f(x)  he  integrable  in  51,  it 
is  necessary  and  sufficient  that  for  any  fair  of  positive  numbers  co, 
cr,  there  exists  a  division  D  of  21,  such  that  the  sum  of  the  subinter- 
vals*  of  D  in  which  the  oscillation  of  f(x)  is  >&),  is  <o-. 

It  is  necessary.  For,  by  497,  there  exists  a  division  D  for  which 
12^/  is  as  small  as  we  please,  and  therefore 

flof=  2g)^S^  <  coo-.  (1 

Let  the  intervals  of  D  for  which  the  oscillation  of  /  is  ^  to,  be 
denoted  by  i)^,  those  for  which  the  oscillation  is  <  co,  by  d^. 

This,  with  lY  gives  „  ,. 

^    °  (oa-  >  cozJJ^. 

Hence  ^  ^ 

It  is  sufficient.     For,  having  taken  e  >  0  small  at  pleasure,  take 

O"  =  ■,      CO  ^  ,  (z 

2{M-my  2(b-ay  ^ 

where 

M=  Max  /,  m  =  Min  /,         in  51. 

*  For  brevity,  instead  of  sum  of  the  lengths  of  the  subintervals. 


344  INTEGRATION 

Then,  by  hypothesis,  there  exists  a  division  I)  for  which 

SD,  <  a. 
This,  with  2),  gives 

<  (M-  m)(T  +  (o(h  —  a)  =  I  +  i  =  e. 
There  is,  therefore,  at  least  one  division  D  for  which 

Then,  by  497,  /  is  integrable  in  %. 

Classes  of  Limited  Integrahle  Functions 

499.  If  f(x)  is  continuous  in  the  interval  51,  it  is  integrahle  in  51. 

For,  since  /  is  continuous  in  5t,  it  is  uniformly  continuous. 
Hence,  by  353,  we  can  divide  51  into  subintervals  of  length  S>0, 
such  that  the  oscillation  of  /  in  each  interval  is  <  &>,  an  arbitrarily 
small  positive  number.  There  is  thus  no  subinterval  in  which 
the  oscillation  >&>. 

Therefore,  by  498,  /  is  integrahle  in  51- 

500.  If  f(x')  is  limited  in  the  interval  51  =  (a,  5)  and  has  only  a 
finite  number  of  points  of  discontinuity  a^,  a^,  ■••  a^,  it  is  integrahle 
in  5t. 

Let  CD,  (T  be  any  pair  of  positive  numbers.  On  either  side  of  the 
points  a^  mark  the  points  a'^,  a'J ,  k=1,  2, 

•  ••  s,  as  in  the  figure;  but  such  that  the     '    ',  ' — 4? irH — 1  h    J 

total  length  of  these  little  intervals  is  <(t. 

Since  /  is  continuous  in  (a,  aj),  we  can  divide  it  into  subinter- 
vals such  that  the  oscillation  of  /  in  each  of  them  is  <  *». 

The  same  is  true  of  the  intervals  (a",  a'2),  {a'^l ^  ag),  ••• 

But  this  set  of  subintervals  in  51  gives  a  division  of  51  for  which 
the  sum  of  the  intervals  in  which  the  oscillation  is  >«  is  <o-. 
Hence,  by  498,  /  is  integrahle  in  51. 


CLASSES   OF   LIMITED   INTEGRABLE   FUNCTIONS         345 

501.  1.  In  263  we  saw  there  were  point  aggregates  having 
derivatives  (not  0  of  course)  of  every  order.  This  leads  us  to 
divide  point  aggregates  into  two  classes  or  species.  The  first 
embraces  all  point  aggregates  whose  derivatives  after  some  order 
vanish.  The  second  embraces  aggregates  having  non-zero  deriva- 
tives of  every  order. 

2.  Let  f(x)  he  limited  in  the  interval  2(.  If  its  points  of  discon- 
tinuity/ form  an  aggregate  A  of  the  first  species,  f  is  integrable  in  21. 

We  note  that  the  aggregate  A  in  500  is  of  order  0. 

We  prove  the  above  theorem  by  complete  induction.  Let  us 
assume  therefore  that/ is  integrable  when  A  is  of  order  n  —  1,  and 
show  /  is  integrable  when  A  is  of  order  n. 

Since  A  is  of  order  n,  A^"^  embraces  only  a  finite  number  of 
points,  by  265.     Call  these 

As  in  500,  we  can  inclose  them  in  little  intervals  («{,  aj')  ••• 
The  points  of  discontinuity  in  the  intervals 

2li=(a,  aO,    %,=  (ia[',a',),   •.• 

form  an  aggregate  of  order  w  —  1.     Hence,  by  hypothesis,  f{x)  is 
integrable  in  ^Ij,  Slg'  "" 

Then  each  interval  21^  can  be  divided  into  subintervals  by  498, 
so  that  the  sum  of  the  intervals  in  which  the  oscillation  of  /  is  >  to 
is  as  small  as  we  please.  As  in  500,  the  totality  of  these  little  sub- 
intervals  furnishes  a  division  of  21  for  which  the  sum  of  the  inter- 
vals in  which  the  oscillation  is  >cd  is  <<t,  an  arbitrarily  small 
number.     Then,  by  498,  f(^x)  is  integrable  in  21. 

502.  Let  f(x)  he  limited  and  monotone  in  21 ;  then  f(x)  is  inte- 
grahle  in  21. 

If  f^pc)  is  constant,  the  theorem  is  obvious.  We  may  therefore 
exclude  this  case.  We  show  that  for  each  €>0  there  exists  a 
division  D  for  which 

Then,  by  497,  /  is  integrable. 


346 


INTEGRATION 


To  fix  the  ideas,  suppose  f(x)  is  increasing.     Let  us  divide  21 
into  equal  intervals  of  length 

e 


8< 


Then 


/W-/(«) 


(1 


i:ii>/=s[i/K)-/(«)i+i/(s)-/K)i+  -  +i/(S)-/K-i)n 

<e,byl). 


503.    Example. 
For 


let 


Let 


2"+!  2" 

/(^)=|.-         «  =  0,  1,2,  3, 
/(0)  =  0. 


Here /is  monotone  increasing  and  limited  in  the  interval  2t  =  (0, 1).  It  is  there- 
fore integrable  in  21.  It  has  an  infinite  number  of  points  of  discontinuity,  viz.  the 
points 

X  =  —  •        n  =  1,  2,  ••• 
2" 


Properties  of  Integrable  Functions 
504.    Iffi(^x),  '"  fsij^^  ^^^  limited  and  integrable  in  the  interval  %. 

Then  -^(2^)  =  (^if\  H H  '^sfs-:         ^'^  constants^ 

is  integrable  in  21,  and 

f^Fdx  =  c,f^f,dx  +  ...+  c,£fdx.  0 

For, 

Passing  to  the  limit,  we  get  1). 


505.    If  f(x^,  g(x)  are  limited  and  integrable  in  21, 
is  integrable  in  21. 


f 


PROPERTIES   OF   INTEGRABLE   FUNCTIONS  347 

1.  Suppose  first  that /(a;),  g(x)  are  >0  and  <  Jf  in  31-     Let  D 
be  any  division  of  %.     Let  h^  be  one  of  the  subintervals.     Let 

0,.   0^   0, 

be  the  oscillations  of  A(a:),  /(a;),  g(^x)  in  S^. 
Let 

F,  /, 

be  the  maximum  and  minimum  of  fQx)  and  g(ix)  respectively  in 

Then 

0,^Fa-fg  =  FCa-g)  +  g(:F-n 

=  FO,-\-gOf<MO,  +  MO,. 
Hence 

£loh  <3r2  0/^  +  Ml,  0,8^  =  Mn^f  +  Mnj)g, 

Since  for  S  =  0, 

lim  D,of=  0,  lim  fl^,^  =  0, 
we  have 

lim  11^^  =  0. 

Hence,  by  495,  h{x)  is  integrable  in  21. 

2.  If  f(x),  g(x)  are  not  positive,  we  can  add  the  positive  num- 
bers a,  /3  to  them  so  that 

/i(a;)  =/(a;)  +  «,  ^'iC^)  =  5'(^)  +  ^ 

are  positive,  and  by  504,  integrable. 
Then 

f^ix)g^(x)  =f(x)g(x}  +  agix)  +  ^f(x)  +  a/9. 

Hence 

f(x)g(x)  =U(x)g^{x)  -  ag(x)  -  ^f(x)  -  a/3.  (1 

Each  term  on  the  right  side  of  1)  is  integrable  by  what  precedes 
Hence  f(x)g(x)  is. 


348  INTEGRATION 

506.  In  the  interval  31  let 

A<fiix)<B, 

where  A,  B  are  both  positive  or  both  negative  numbers.     If  f(x)  is 
integrable  in  21,  so  is 

To  fix  the  ideas,  suppose  A,  B  are  positive.     Using  the  notation 

of  505, 

_1       1  _F-f     F-f  _  0. 
■       '~f     F~    fF    '^    A^    ~J2' 
Hence 

As  /(.r)  is  integrable, 

limn^/=0. 

6=0     ■ 

Hence  gQx)  is  also  integrable,  by  1)  and  495. 

507.  If  f(x)  is  limited  and  integrable  in  21,  so  is 

^(a^)  =1/(^)1  • 
For,  using  the  notation  of  505, 

obviously.     Therefore 

o<fi^^<n^/. 

As 

lim  Q^jjf  =  0, 

ff(x)  is  integrable  by  495. 

508.  In  501,  503  we  have  met  with  integrable  functions  which 
have  an  infinite  number  of  discontinuities  in  2t.  There  is,  how- 
ever, a  limit  to  the  discontinuity  of  a  function  beyond  which  it 
ceases  to  be  integrable,  viz. : 


FU^^CTIONS   WITH   LIMITED   VARIATION  349 

If  the  limited  function  f(x)  is  integrable  in  the  interval  21,  there 
are  an  infinity  of  points  in  any  partial  interval  ^  of  %^  at  which  f 
is  continuous. 

Let  &)j>ft)2>  ■••  be  positive  and  =0.  By  498  there  exists  a 
division  such  that  the  sura  of  the  intervals  in  which  the  oscilla- 
tion of  /is  >&)j  is  <(T.  Thus,  if  a-  be  taken  less  than  ^,  there  is 
at  least  one  subinterval  within  ^,  call  it  ^j,  in  which  the  oscillation 
is  <&)j.  Similarly,  there  is  an  interval  ^^  within  :35ii  hi  which  the 
oscillation  of/  is  <co^.  Continuing  this  way,  we  can  get  a  sequence 
of  intervals 

each  contained  within  the  preceding,  whose  lengths  =  0.  Then, 
by  127,  2,  the  sequence  1)  defines  a  point  c  within  ^,  such  that  for 
every  point  x  in  Dip), 

\f(x)  —  f(c')  I  <  ft).         ft)  arbitrarily  small. 

Thus  S  contains  at  least  one  point  c,  at  which /(a;)  is  continuous. 
It  therefore  contains  an  infinity  of  such  points. 

Functions  with  Limited  Variation 

509.  An  important  class  of  limited  integrable  functions  is 
formed  by  functions  with  limited  variation,  which  we  now  consider. 

Let /(a;)  be  defined  over  the  interval  51  =  (a,  J). 
Let  i>  be  a  division  of  21  of  norm  8;  let  Sj,  h^.,  •••  be  the  sub- 
intervals  of  21  corresponding  to  this  division. 

Let  ft)^  denote  the  oscillation  of /(a;)  in  h^.     Let  us  form  the  sum 

ft)  =  Max  (O]) 

is  finite  for  all  possible  divisions  i),  we  say  f(x)  is  a  function  with 
limited  variation.,  or  that  fQc)  has  a  limited  variation  in  21. 

We  call  5  the  variation  of  f(x)  in  21.  If  S  is  infinite,  we  say 
f(x)  has  unlimited  variation  in  21-  If  f(x)  is  unlimited  in  21,  it 
cannot  be  a  function  with  limited  variation  in  21- 


350 


INTEGRATION 


510.    Let  i)  =  (aja2"0  be  a  division  of  31.     Let  us  form  a  new 
division  A  by  interpolating  a  point  a  between  a^_i,  a^. 
Then  h^  falls  into  two  intervals  8[,  h['  in  A. 


Let 


M,,  M[,  M[' 


be  the  maximum  of  /  in       S^,     8[,     S/', 


and  let 


w,,    m[,    m[', 


be  the  minimum  of  /  in  these  intervals. 
Then  the  term 

in  Q)^  is  replaced  by 

c»[  +  ft,;'  =  (M[  -  m[)  +  (iM['  -  ml') 

in  ft)^.     Now  at  least  one  of  the  M[,  M['  equals  M^ ;  and  at  least 
one  of  the  m[',  m[  equals  m^.     To  fix  the  ideas,  let 


Then 


M[  =  M,,    m['  =  w, 


ft,^  =  (ft,[  +  ft,[')  _  ft,^  =  ilif  [' _  ^[  ^  0. 


(1 


511.    1.  Let  fQc)  he  a  limited  monotone  function  in  21.      It  has 
limited  variation  in  21. 

To  fix  the  ideas,  let  it  be  monotone  increasing.     Then 

«/>  =  l/(«i)  -/(«)!  +  l/(«2)  -/K)l  +  -  +  \f(h^  -/(«„-i)l 

Thus,  whatever  division  D  is  employed,  fOp  has  the  same  value. 

Hence 

S=/(J)-/(a). 


2.  Let 


2l  =  2ti  +  2(2-+2l^. 


Letf(x)  he  monotone  and  limited  in  each  interval  21^.  Thenf  has 
limited  variation  in  21. 

For,  we  get  the  maximum  value  of  co^  when  D  embraces  all  the 
end  points  of  the  intervals  2l«.  In  fact,  let  i>  be  a  division  which 
does  not  include  one  of  these  end  points,  say  a,  which  lies  in  the 


FUNCTIONS   WITH   LIMITED   VARIATION 


351 


interval  5,.     Let  A  be  a  division  formed  by  adding  a  to  D.     Then, 
by  510,  1), 

If,  on  the  other  hand,  i)  is  a  division  including  all  the  end 
points  of  2lj,  Slg,  •••  we  cannot  increase  (o^  by  adding  other  points 
to  i>,  as  we  saw  in  1.  Thus  the  variation  oif(x)  in  %  is  the  sum 
of  the  variations  in  each  5l«.  As  these  latter  are  j&nite  by  1,  /  is 
of  limited  variation  in  %. 


.  -^ 


512.  1.  It  is  easy  to  construct  functions  having  an  infinite 
number  of  oscillations  in  21,  which  are  of  limited  or  unlimited 
variation. 

Let  J  >  1,  and  in  51  =  (0,  5)  take  the 
aggregate  1     i    1     i 

J^i     2'     3'     t' 

Let  the  line  OL  make  the  angle  45° 
with  the  a;-axis.     Between  each  pair  of    6 
points  ^ 

take  a  point  a^. 

Let /(a;)  have  the  graph  formed  of  the  heavy  lines  in  the  figure. 

^''  n    (^  1  ^\ 

X*  =-,«„_!,    -,    •••    «!,  11- 

\n  11  — 1  J 

Then  All  1\ 

„„=  2(1  +  1  +  1+...  +  !). 

As  we  shall  see  later,  the  limit  of  the  expression  on  the  right  is 
infinite. 

Hence  f(x)  is  of  unlimited  variation. 

2.  To  form  a  function  having  limited 
variation,  take  the  parabola, 

instead  of   the  right  line  OL  in  the 
last  example. 


Ax)=o 


352  INTEGRATION 

Then  /        1       1  1 

«.=  2(l  +  -  +  -  +  ...+  l 

But  as  we  shall  see,  the  limit  of  the  right  side  is  here  finite. 
Hence /(a;)  is  of  limited  variation. 

3.  Similar  considerations  show  that 

y  =  x  sin  - ,  a;  ^  0  ;  y  —  ^  for  re  =  0 

^  X 

has  unlimited  variation  in  (0,  6)  ;  while 

y  =  x^  sin  -,  a; ^  0  ;         y  =  ^  for  x  =  0 
has  limited  variation  in  (0,  J). 

513.  If  f(x)  has  limited  variation  in  21,  it  is  integrahle  in  21. 

We  apply  the  criterion  of  498.  Let  then,  oo,  <t  be  an  arbitrary 
pair  of  positive  numbers.  Let  D  be  a  division  of  norm  h.  Let 
V  be  the  number  of  subintervals  in  which  the  oscillation  of  /(a;) 
is  >  eo.     Then,  for  any  division  whatever, 

where  a  is  the  total  variation  of  /  in  21- 
Let 

then 

v<p. 

Let  us  take  S  <  - .     Then  the  sum  of  the  intervals  in  which  the 

oscillation  of/is  >»  is 

<.vB<pS<C  <T. 

Content  of  Point  Aggregates 

514.  1.  Let  A  be  a  point  aggregate  lying  in  the  interval  21. 
For  example,  A  may  be  the  interval  21  itself.  Or  it  may  consist 
of  a  certain  number  of  partial  intervals  of  21.  Or  it  may  embrace 
an  infinity  of  subintervals  with  or  without  their  end  points  after 


CONTENT  OF  POINT  AGGREGATES         363 

the  manner  of  Ex.   7,  8  in  271.     Or  it  may  consist  of  a  mixed 
system  of  intervals  and  points  not  forming  intervals. 
Let  us  effect  a  division  i)  of  21  into  subintervals 

as  heretofore. 

Let  Si,    B',,    .-.  (1 

be  those  intervals  of  D  in  which  points  of  A  fall.     Let 

S[',   B',',    ...  (2 

be  those  intervals  of  i),  all  of  whose  points  lie  in  A. 

A  =  lim  1.8',,         A  =  lim  ^B'J 

S=o  6=0 

exist  and  are  finite. 

In  fact,  let  us  introduce  an  auxiliary  function  /(a;)  whose  value 
is  0  in  21,  except  at  the  points  of  A,  where  its  value  is  1. 

Then  evidently,  using  the  notation  and  results  of  490,  491, 

|S,  =  ^MA  =  S„ 
since  M,  =  1  if  S^  is  in  1),  but  is  otherwise  0. 

since  m,  =  l  if  B,  is  in  2),  but  is  otherwise  0. 

Hence  _     _ 

A  =  S\        A  =  S. 

2.  The  numbers  A,  A  are  called  the  upper  and  lower  content  of  A. 

When  A  =  A, 

their  common  value  is  called  the  content  of  A.     We  denote  it  by 

Cont  A, 

oj'  when  no  confusion  can  arise,  by  A. 

The  upper  and  lower  contents  may  be  denoted  by 


A  =  Cont  A,         A  =  Cont  A. 
A  limited  point  aggregate  having  content  is  said  to  be  measurable. 


354  INTEGRATION 

515.  Let  ^  he  a  partial  aggregate  of  31.  Let  (5  =  21  — ^  he  the 
complementary  aggregate.     If  21  and  ^  are  measurable.^  so  is  d,  and 

Cont  (5  =  Cont  21  -  Cont  ^.  (1 

For,  let  i)  be  a  division  of  norm  8.  Let  §  be  the  frontier  of  ^. 
Let  21^),  ^^,  S/>,  ^^23  be  the  sura  of  the  intervals  of  D  containing 
points  of  21,  ^,  S,  5,  respectively.  Let  SC^,  ;^^,  ^^^  be  the  sum  of 
the  intervals  which  lie  wholly  in  21,  ^,  (5,  respectively. 

Then 

i^<^^  +  e,,<f^  +  f^,  (2 

since  some  of  the  intervals  of  D  may  contain  both  points  of  ^ 
and  (5,  and  are  therefore  counted  twice  on  the  middle  member  of  2). 
Similarly, 

Passing  to  the  limit,  S  =  0,  in  2),  3),  we  get 

I<«  +  g<2t, 
2l>^  +  f  >2l. 

But  I  =  2t  =  Cont2l,        B  =  ;^=Cont«. 

Hence 

i  =  ^=  Cont  21- Cont  «S, 
which  gives  1). 

516.  1.  By  the  aid  of  the  auxiliary  function /(a;)  introduced  in 
514,  1,  the  criteria  for  integrability  which  we  deduced  in  495,  497 
give  at  once  criteria  that  A  have  a  content.     Thus  495  gives  : 

In  order  that  A  have  content,  it  is  necessary  and  sufficient  that  the 
sum  of  those  intervals  containing  hoth  points  of  A  and  points  not  in 
A,  converge  to  zero,  as  the  norm  8  of  D  converges  to  zero. 

2.  From  497  we  have : 

In  order  that  A  have  content,  it  is  necessary  and  sufficient  that  for 
each  positive  numher  e  there  exists  a  division  D  of  %,  such  that  the 
sum  of  the  intervals  in  which  hoth  points  of  A  and  not  of  A  occur, 
is  <e. 


CONTENT   OF   POINT   AGGREGATES  355 


EXAMPLES 

517.   1.  A  =  rational  numbers  in  31  =  (a,  b). 
Here 


A  =  b~a,     A  =  0. 

^                                             A  =  0,  1,  ^,  i,  i,  ...           4*5 

^1     1 

Let  e  >  0  be  arbitrarily  small.                                    "         ^       2 

1 

Let  us  define  the  division  D  as  follows.     Inclose  each  of  the  points 

1     1     1     ..        1 

'    2'   3'         n-l 
within  intervals  of  length 

3^' 
where 

e 

The  remaining  points  of  A  fall  in  the  interval 

("■i-^y  »=r 

Then  the  sum  of  the  intervals  containing  both  points  in  A  and  not  in  A  is  <e. 

Hence,  by  516,  2,  A  has  a  content. 

As  obviously  A  —  0 

we  have  z-c     *  a      n 

Cont  A  =  0. 

If  E  is  the  complement  of  A  in  21, 

Cont  E  =  Cont  31  =  (6  -  a), 
by  515. 

518.  1.  A  point  aggregate  of  content  zero  is  called  discrete. 
Such  an  aggregate  is  given  in  517,  Ex.  2. 
Every  limited  point  aggregate  of  the  first  species  is  discrete. 
The  reasoning  is  perfectly  analogous  to  that  of  501. 

2.  Let  f(x^  he  limited  in  the  interval  21.     If  the  points  of  discon- 
tinuity of  f(x)  form  a  discrete  aggregate.,  /(»)  is  integrahle  in  %. 

This  follows  at  once  from  497. 

3.  Let  y=f(x)  he  univariant  in  %.     Let 
^  <M,   if  Ax<d. 


356  INTEGRATION 

Let  A  he  a  discrete  aggregate  in  21.  The  image  E  of  L  is  also 
discrete. 

Let  us  effect  a  division  of  21  of  norm  8  <  d. 

Let  Sj,  ^2,  •••  be  the  subintervals  containing  points  of  A.  Let 
7;j,  t]^,  •■•  be  the  corresponding  intervals  on  the  «/-axis.  Then,  by 
hypothesis, 

■^^"^  ^v.<M^K-  (1 

But  A  being  discrete,  we  can  take  S  so  small  that  the  right  side 
of  1)  is  <  e. 

Hence  U  is  discrete. 


Generalized  Definition  of  an  Integral 

519.  Up  to  the  present  we  have  supposed  that  the  integrand  f(oc) 
is  defined  for  all  the  points  of  the  interval  21=  (a,  5).  By  employ- 
ing the  results  of  the  last  articles,  we  can  generalize  as  follows  : 

Let  ^  be  a  measurable  aggregate  in  21,  and  let  f(x)  be  a  limited 
function  defined  over  ^.     Let  us  effect  a  division 

D{a-^^,  ^2'  ■■■) 
of  21  of  norm  8. 

Those  intervals        y^  ^     ^  ^  ^-, 

all  of  whose  points  lie  in  ^,  form  a  system  which  we  denote  by  D^ 
The  lengths  of  the  intervals  1)  we  denote  by  SJ,  S^,  •••;  while 

111  I21  •••  are  points  taken  at  pleasure,  one  in  each  interval  of  1). 
Let  us  build  the  sum 

•"1 

If  as  S  =  0,  Jg  converges  to  one  and  the  same  value,  however  the 
divisions  D  and  the  |'s  be  chosen,  we  call  this  common  limit  the 
integral  of  /(a;)  over  Sd,  and  denote  it  by 

We  say  in  this  case  that  f(x^  is  integrable  with  respect  to  ^, 


GENERALIZED   DEFINITION   OF   AN    INTEGRAL  357 

520.  Let  f(x)  he  a  limited  function  defined  over  a  limited  point 
aggregate  ^  of  content  zero.      Then  f(x)  is  integrable  over  ^^  and 

f^fCxydx^o. 

T,pf 

have  the  same  meaning  as  in  519. 

Since  /  is  limited,  let 

|/(^)|<i»f. 
Then 

Since  ^  is  of  content  zero, 

lim  2S;;  =  0. 

5=0 

Hence 

lim  Jg  =  0, 

5=0 

which  proves  the  theorem. 

521.  1.  Let  f(cc)  he  a  limited  integrahle  function  defined  over  the 
measurable  aggregate  ^.     Let  the  interval  51  =  (a,  5)  contain  ^. 

g(x)  =  fQc),         in^; 

=  0,        for  points  of  51  not  in  SQ. 

Then  g(x)  is  integrahle  in  (a,  6),  and 

y  g{x)dx  =  ^^f{x)dx.  (1 

Let  us  effect  the  division  2)  of  51  as  in  519. 

As  before,  let  L^  be  the  system  of  intervals  lying  in  ^.  Let  D^ 
be  the  system  of  intervals  containing  no  point  of  ^.  Let  A  be  the 
system  of  intervals  containing  both  points  of  .33  and  points  not  in  ^. 

We  build  now  the  sum 

with  reference  to  the  interval  51. 


358  INTEGRATION 

Since 

we  have 

J,  =  ^giLA  +  l^i^JK  +  f ^(L)S..  (2 

Since  g(:x)—  0  in  i>2,  the  second  sum  in  2)  is  0.     Since 
in  _Z>j,  we  have  now 

Now,  by  hypothesis,  /  is  integrable  in  ^,  and 

On  the  other  hand, 

lim|.9(fj8,  =  0, 

6=0    ^ 

by  516  and  520. 

Hence,  passing  to  the  limit,  S=  0,  in  3),  we  have  1). 

2.  The  reasoning  in  1  gives  as  corollary : 

If  f(x)  is  limited  m  ^,  and  g{x)  is  integrable  in  21,  then  f(x)  is 
integrable  in  ^,  and 

f  gQx)dx=  i  f(x)dx. 

This  is  at  once  evident,  on  passing  to  the  limit  in  3). 

3.  Let  /(a:)  be  limited  in  %  =  (a,  b).  Let  A  be  a  discrete  aggre- 
gate in  51.  Let  ^  =  2t  —  A.  Let  /(a;)  be  integrable  in  ^.  Then 
f(x)  is  integrable  in  21,  and 


Jfdx  =  I    fdx. 
a  c/SB' 

The  demonstration  is  similar  to  1,  omitting  the  system  D^. 

522.  1.  Let  f(x~)  be  a  limited  integrable  function  ivith  respect  to 
the  measurable  aggregate  iB,  lying  in  the  interval  2l  =  («,  ^).  Let 
D  =  (ay,  ^2'  ■■■  ^n-\)  ^^  ^  division  of  %  of  norm  8.     Let 


GENERALIZED    DEFINITION   OF   AN   INTEGRAL  359 

he  resulting  intervals  formed  of  one  or  several  contiguous  intervals  of 
-Z),  lying  in  ^.      Then 

( fdx  =  \imXrydx.  (1 

Let  us  introduce  the  auxiliary  function  g{x)  of  521.     Then,  by 
521, 


Now  the  division  D  breaks  51  up  into  the  intervals 
(a,  aj),  (ap  a^,  (a^,  ag),  •••  (a„_i,  5). 
Letting  i>j,  i>2,  A  have  the  same  meaning  as  in  521,  we  have 

f  gdx  =  T  f'^V^-^'  =  2  f'^  V<^-^  +  2  I    '^V^^;  +  2  (    '^V^2;. 

»/a  V*^«'  ^^^''i  ^"^"t  A  ^«i 

But 
while 

2  f'^V^^ = o» 

i),  »/  a^ 

since  ^  =  0  in  the  intervals  of  D^. 
Thus, 

j^dx = xjydx + 2X"'^'^'^'^-  ^^ 

Now,  if 
we  have,  by  489,  4, 

1 2  C'^'gdx  <  if  2  («^+i  - «.)  =  ^^' 


But  since  ^  is  measurable. 


lim  A  =  0. 


Hence  the  second  terra   on  the  right  of  2)  has  the  limit  0. 
Hence,  passing  to  the  limit  in  2),  we  get  1). 


360  INTEGRATION 

2.  The  preceding  reasoning  gives  the  corollary  : 

If  f(x)  is  limited  in  ^  and  integrahle  in  each  of  the  intervals 
(««'  /5«)'  ^^^  2  I    y^^  *^  convergent  as  S  =  0,  then 

Jfdx  =  lini  V  (    fdx. 

This  is  evident  on  passing  to  the  limit  in  2). 

3.  Letf(x)  he  limited  in  31=  (a,  5).     Let  A  be  a  discrete  aggre- 
gate in  31.     Let^  =  %-  A.      Then 

Jfdx  =  lim  ^  I    fdx^ 

provided  the  limit  on  the  right  is  finite. 

For,  by  2, 

lim  V  I    "/(ia;  =   Cfdx. 

S=0      ^»^'='k  '^S 

But,  by  521,  3, 

Jfdx=   i  fdx. 


CHAPTER   XIII 
PROPER  INTEGRALS 

First  Properties 

523.  In  the  last  chapter  the  integrand  f(x)^  as  well  as  the 
interval  of  integration  21,  were  limited.  Integrals  for  which  this 
is  the  case  are  called  proper  integrals,  in  contradistinction  to  those 
in  which  either  f(x)  or  21  is  unlimited.  These  latter  are  called 
improper  integrals. 

In  this  chapter  we  consider  only  proper  integrals.  We  wish  to 
establish  their  more  elementary  properties. 

In  21  =  (a,  5),  we  shall  take  a<6,  unless  the  contrary  is  stated. 
All  the  functions  employed  as  integrands  are  supposed  to  be  limited 
and  integrable  in  21- 

524.  For  the  sake  of  completeness,  we  begin  by  stating  the  three 
following  properties  already  established  respectively  in  489,  2; 
489,  4 ;  504,  viz.  : 

f{x)dx=— ^^f{x)dx,  a^b.  (1 

\f^f{^x)dx\<M\h-a\,        a^b,  (2 

|/(a;)|  being  <Min  the  interval  (a,  6). 

I Hfi  -^ ^-  (^sfs\d^  =  ^1  I  fidx H VcA  f,dx,      a^b.    (3 

525.  Let  a,  b,  c,  be  three  points  in  any  order.      Then 

Xb  /•c  fb 

fdx  =  ^Jdx  +  jjdx.  (1 

361 


362  PROPER  INTEGRALS 

Suppose  first  that  a<e  <h.     Since 

is  the  same,  whatever  system  of  division  D  we  choose,  let  us  coii' 

sider  only  such  divisions  in  which  c  enters.     The  points  of  I)  which 

fall  in  (a,  c),  let  us  call  D^ ;   those  falling  in  (c,  6),  call  J)^. 

Then 

iA  =  2A  +  2A.  (2 

Now  f(x)  being  integrable  in  51,  is  integrable  in  (a,  c)  and 
(c,  J),  by  496. 

Hence,  passing  to  the  limit  in  2),  we  get  1). 

The  theorem  is  now  readily  established  for  any  other  order  of 
a,  6,  c. 

526.    1.  In%  =  {a,  6),  let 

m<f(x)<M. 
Then 


m(b-a)<Jfdx<M(b-a).  (1 

or  m{b  —  a)  <J^  <  M(h  —  a). 

Passing  to  the  limit,  8  =  0,  we  get  1). 
2.  In  %  let  f{x)^g{x).      Then 

f/d.>f^gdx.  (2 

For, 

hCx-)=f(x}-g(x')%0,         in  SI. 

Hence,  by  1), 

f^hdx=:f^fdx-f^gdxsO, 
which  gives  2). 


FIRST   PROPERTIES  363 

527.  1.  We  saw  in  508  that,  if /(a:)  is  integrable  in  21,  it  must 
have  points  of  continuity  c,  in  any  subinterval  of  21.  This  fact 
leads  us  to  state  the  following  theorem : 

Letf(x)^^  in  21.     Iffis  continuous  at  c,  andf(^c')>0,  then 


//' 


'dx>0. 


To  fix  the  ideas,  suppose  c  is  an  inner  point.     Then  by  351,  2, 
there  exists  an  interval  (c',  e")  about  c,  in  which /(a;)  >  p  >  0. 
Hence  by  526, 


But 


£fdx^O,   £"fdx^p(c"-c'}  =  <T,    £fdx%0. 


2.  As  corollary  we  have  : 

Letf{x)  ^g(x)  hi  21.      If  at  a  point  c  of  continuity  of  f  and  g^ 

/(O  >^(0i  then 

I    fdx>  I  qdx. 

3.  By  means  of  the  preceding  inequalities,  we  can  often  estimate 
approximatel}^  the  value  of  an  integral  with  little  labor,  as  the 
following  examples  show. 

Ex.1.  .b<P—^ — <.5236.         ?i>2.  (1 

For,  if  0  <  X  <  1,  1  1 


Hence  * 

which  gives  1). 
Ex.  2. 


Vl  —  X"      Vl  —  x2 

I    dx<,\     — ^^ri^r:  <  1     — =  =  arc  sin  i  =  - , 

Jo  J     Vl  _  rn      Jo    VI  _  .r.2  6 


xe-'^<  r'e-«'dtt<arctgx,        x>0.  (2 

For  by  413,  2,  2 

€"  =  1  +  z+—e^',        0<^<1, 

*  In  order  to  illustrate  these  and  a  few  immediately  following  theorems  we  assume  the 
elementary  properties  of  indefinite  integrals,  which  are  treated  in  536  seq. 


364  PROPER  INTEGRALS 


Hence  e^>l  +  s,        z>0. 

Thus  _   ^2  ^  „_„2  ^1  -c  n   ^       ^^ 

e-»:  <e-»  <-—; — -,        ifO<«<x. 
1  +  u^ 
Hence,  x  being  a  constant, 


0  Jo  Jo  1 


(itt 


-  +  m2  ' 

which  gives  2). 

528.  1.  Let  f(x)  he  limited  and  integrable  in  21;  then  \f(jic)\  is 
integrable  in  21,  and 

I  ( fdx\<  f  \f\dx.  n 

In  the  first  place,  |/|  is  integrable  by  507. 
On  the  other  hand. 

The  relation  1)  follows  now  by  526. 

2.   The  reader  should  guard  against  the  error  of  assuming  that 
f{x)  is  necessarily  integrable  in  21  if  \f(x')  \  is  integrable  in  21- 
For  example,  let 

f(x)  =  1  for  rational  x  in  21, 

=  —  1  for  irrational  a;  in  21. 

1/(2;)  I  =  1,  for  every  x  in  21, 

and  is  therefore  integrable  in  21-     But /(a;)  is  obviously  not  inte- 
grable in  2t. 

529.  1.  /ri2t  =  (a,  5),  let 

m<g(x)<M.  (1 

When  not  zero,  let  f(x)  be  positive.     Then 

mjjdxKJjgdxKMJjd^.  (2 

For,  multiplying  1)  by  /(a:),  we  have 

mf<f9<Mf. 
We  have  now  2)  by  526. 


FIRST   PROPERTIES  365 

2.   In  %  let  f(x)^Q.     Let  m<g{x)<M.     At  a  point  c  of  con 
tinuity  of  f(x)^  ^(2^)9  ^^t 

f(ic^>0,         m<gCc}<M. 

Then  C  r  C 

m  I   fdx  <       fqdx  <  M  I  fdx. 

We  have  only  to  apply  527,  1  to  the  functions 

QM-g^f  and    (^-m)/. 
^^''^-  arc  sin  x<  T—        ^^  ^  arc  sin  x  ^g 

•^^     Vl  -  X2  .  1  -  XX2         Vl  -  X 

where  0<\<1,    0<x<l. 

For,  1  1 

1<  <- 


Vl  -  Xx2      Vl  -\ 
Hence 

J'»    (7x     ^  r* rfx ^     1     r»= 


(?X 


C2V1  -  XX2         Vl-X*'"     Vl  -  X2 

which  gives  3). 

F^^-  2-  .333  <  r  ^  x^e-'fZx  <  .  907.  (4 

For,  if  0<x<l, 

1  <  e^'  <  e. 
Hence 

I    orMx  <  \    x^e'^'clx  <  e  |    x^dx, 

which  gives  4). 

530  1.  Let  f(x}  =  f/(x')  in  21,  except  fur  the  points  of  a  discrete 
aggregate  A ;  then  f  g  being  limited  and  integrable. 

Then  h  —  0,  except  for  the  points  of  A. 

Let  D  be  a  division  of  21.  Let  D^  embrace  those  intervals  con- 
taining no  points  of  A.  Let  D^  embrace  the  other  intervals  of  D. 
Then  ,    ,     ^ 

SKIA  =  s  +  s  =  S. 

D  D^        Z),        2), 


366  PROPER  INTEGRALS 

Let  now  3=0.     The  limit  of  the  left  side  is 

Xhdx. 
Since  A  is  discrete,  the  limit  of  the  right  side  is  0  by  520.     Hence 

Xhdx  =  0, 


which  proves  1). 

2.  As  a  corollary  we  have : 
In  the  integral 


=  //<'- 


we  may  change  at  pleasure  the  value  of  f(x)  at  the  'points  of  an 
arbitrary  discrete  aggregate^  without  changing  the  value  of  «/,  pro- 
vided the  new  integrand  is  also  limited  in  %. 

First  Theorefn  of  the  Mean 

531.    Let  f(x),  9(3^  ^^   limited   and   integrable   in   %=(a,  5). 
When  not  0,  let  f(x)  he  positive. 
Then 

fgdx=G-jJdx,         a^h,  (1 

where  Gr  =  Mean  g(x)i         in  21. 

For,  by  definition,  268,  4, 

m<Cr<M, 

where  m  and  M  are  respectively  the  minimum  and  maximum  of 
g(x^  in  21.     Also,  by  529, 


m 


''  ff^^<  ( fgdx<M  Cfdx,         a<h, 

which  gives  1)  in  this  case.     The  case  oi  a>b  follows  now  at 
once. 

The  above  is  called  the  first  theorem  of  the  mean.     We  give  now 
some  special  cases  of  it. 


FIRST   THEOREM   OF   THE   MEAN  367 

532.  Letf(x)  he  integrable  in  21  =  (a,  h').     Let 

m  =  Mean/(a;),         in  %. 
Then  ^b 

jj(x)dx  =  m(h-a~).         a<h. 

Proved,  by  taking  one  of  the  functions  in  531,  equal  to  1. 

533.  In  the  interval  %  =  (a,  5),  let  f(x)  he  limited^  integrahle, 
and  non  negative.     Let  g(x)  he  continuous.      Theii 

Cfgdx=gQ}  Cfdx,         a<^<h.  (1 

For,  by  531,  setting  (7  =  Mean  ^(2;), 

Jfgdx=  Gr  I  fdx. 
21  ^2t 

But  g(x')  being  continuous,  takes  on  every  value  between  its 
extremes  including  end  values,  while  x  runs  over  21,  by  357.  Hence 
for  some  |  in  21, 

which  proves  1). 

534.  In  the  interval  21  =  (a,  h)  let  f(x)  he  limited.,  integrahle  and 
non  negative.  Let  g(x)  he  continuous.  At  some  point  of  continuity 
of  fix)  let 

m<g(x)<M, 

where  m  =  Min^(a;),  31=  Max^(a;),  in  21. 
Then 


ffadx=gQ)  ffdx.         a<^<h. 
m  Cfdx<  C fgdx<M  C fdx. 


'21 

For,  by  527,  2, 

m  I 

21  -^21 

Hence, 

Cfgdx  =  a  Cfdx,        m<a<M. 


368  PROPER   INTEGRALS 

Now  g(x)  being  continuous  takes  on  its  minimum  m  at  some 
point  a,  and  its  maximum  at  some  point  /3  in  21,  by  357.  More- 
over, at  some  point  |  in  the  interval  (a,  yS),  g(x)  must  take  on  the 
value  G-.  As  g(x)  has  the  values  m  and  M  at  the  end  points  of 
this  interval,  |  must  lie  within  this  interval.     Hence 

a<^<h. 

535.  Letf(x)  he  continuous  in  21=  (a,  5).      Then 

^Jdx  =  (5  -  «)/(!),         a  <  f<  5. 

This  is  a  corollary  of  534,  one  of  the  functions  being  1,  provided 
fix)  is  not  a  constant,  when  the  theorem 
is  obviously  true. 

This    theorem    admits    a   simple   geo- 
metric   interpretation.        It    states   that 

there   is  a  point  |,    a  <  |  <  5,  such  that      \ \ \ 

the  area  of  the  rectangle  ABC D'  is  the 

same'  as  that  of  the  figure  ABCD,  determined  by  the  graph  of /(a;). 

The  Integral  considered  as  a  Function  of  its  Upper  Limit 

536.  Let  f(x)  he  limited  and  integr able  in  %  =  ia,  h).      Then 

F(x)  =   I  f(x)dx^  a,  X  in  21, 

is  a  one-valued  continuous  function  of  x  in  21. 

Since  f(x)  is  integrable  in  2t,  F  has  one,  and  only  one,  value  for 
every  x. 

Let  I/(a:)|<il!f. 

We  have 

Xx+h  nx  nx-yh 

f{x)dx  -  y  f{x)dx  =  J^     f(-x)dx. 

^^"^^^  |A^|  <  M\h\         by  524,  2). 


INTEGRAL   AS   A   FUNCTION   OF   ITS   UPPER   LIMIT       369 
Thus  if  we  take  5.  ^  e 

^<¥' 

^"'"'™  \AF\<e, 

for  every  h,  such  that  x  +  h  falls  in  31,  and 

\h\<S. 
Hence  -F(a;)  is  continuous  in  St. 

537.    1.   Letf(pc)  he  limited  and  integrahle  in  31  =  (a,  5).     Let 

f(^x}dx,         a,  X  in  21. 
If  f  is  continuous  at  x^         ,  ^ 

To  fix  the  ideas,  let 


ax 


a<a<x<x  +  h<.b. 

^^^"^    A  J = j(x + A)  -  j(x)  =  r^'  -  r=  r^' 

=  }m,     by  532, 

where  3JJ  =  Mean  f{x)  in  (2;,  2;  +  A). 
But  since  /(a;)  is  continuous  at  x^ 

for  any  h  <  some  S. 

Hence  At7     ^^    .   , 

Ax 

Passing  to  the  limit,  we  get  1). 

2.   As  a  corollary  we  have : 

Letf(jc)  he  limited  and  integrahle  in  %  =  (a,  5).      Then 

lira-  I       f(x')dx  =  f(^x'),         xin%  (2 

iff  is  continuous  at  the  point  x. 


370  PROPER   INTEGRALS 

538.    1.  In  the  interval  21,  letf(x)  he  a  limited  int eg r able  function. 
Let  F(x')  he  a  one-valued  function  tvhose  derivative  isf(x).      Then 

£fdx  =  Fi^-)  -  #(«).         «,  /3  in  21.  (1 

For,  let  2)  =  (aj,  a^,  •••  a„_i)  be  a  division  of  the  interval  (a,  /8). 
Then  by  the  Law  of  the  Mean, 

-^(«i)-^(«)=/(li)Si, 

Adding  the  equations  2),  \ve  get,  since  terms  on  the  left  cancel 
in  pairs,  ^^^^  _  ^^^^  ^  2/(^JS^.  (3 

Since /(a;)  is  integrable, 

limS/(|jS,=  r%:.. 

5=0  »^» 

Passing  therefore  to  the  limit  in  3),  we  get  1). 

2.   Example.     By  differentiation  we  verify 

„     X  —  X  arctg  (X  tan  x)  "* 

JJx  "  " 


/x  • 


1-X2  1  +  X2tan2x 

for  I X I  ^  1  and  x^(2m  +  l)7r/2.     Hence  by  538,  1, 

C^  dx  _x  —  \  arctg  (X  tan  x)  0<rx<^—  l\l:il 

Jo  H-X2tan2a;~  1-X2  '  2'        'l'^^- 

The  integrand  is  not  defined  for  x  =  ir/2  ;  let  us  therefore  assign  it  the  value  0. 
The  integral  on  the  left  is  continuous,  by  536.     Hence 

f "''  =:  i  lim  r  =  i  lim  ^-^arctg(Xtanx)^ 

^0  x=7r/2  Jo  1  -  X2 

or 

Jo      l  +  xnan2x      2   1 -X2  ~  2  '  1  +  |X|' 

As  we  have  derived  this  formula  we  have  been  obliged  to  assume  |  X  |  ^  1,     It 
is,  however,  valid  even  when  |  X  |  =  1.     For, 

r'^/2        dx  C^r-      o     ■,        rl      ,  1    •    o    T^/2     IT 

\ —  =  \       cos2  X  {?«  =     -  x  +  -  sui  2  «         =  -) 

^0      l  +  tan2a!     h  L2        4  Jo         4 

which  agrees  with  4)  when  |X|  =  1. 


CHANGE   OF   VARIABLE  371 

539.  Criticism.  A  common  form  of  demonstration  of  tlie  pre- 
ceding theorem  is  the  following.     Since 

we  have 

.>0,         S>0,         Z(^+^lziiM=/(^)  +  ,r,         |^|<,, 

for  |A|<S.     The  equations  2)  of  538  are  now  written 
^(ai)-l^(«)=/(«)Ai  +  eA, 

^(/3)  -  ^K-i)  =/(a„_i) A,  +  eA. 
Adding,  we  get 

l^(/3)  -  Fia)  =  tfia:)\  +  teX-  O 

If  6  is  numerically  the  greatest  of  the  e^, 

|SeA|<eSA,  =  e(;e-a),  yS>a. 

It  is  now  assumed  that  e  =  0  with  S.  Hence  passing  to  the  limit, 
S=0,  1)  gives  538,  1). 

The  objection  to  this  demonstration  lies  in  the  tacit  assumption 
that  the  difference  quotient  converges  uniformly  to  the  derivative. 
Cf.  404.  In  other  words,  that  it  is  possible  to  divide  the  interval 
(a,  ^)  into  subintervals  7ij,  h^,  •••  h^  such  that  e^,  e^-,  ■••  e„  are  all 
<  cr,  a  positive  number,  small  at  pleasure.  As  elementary  text- 
books say  nothing  of  uniform  convergence,  the  above  reasoning  is 
incomplete. 

Change  of  Variable 

540.  1.  Let  f(x)  he  limited  and  integrahle  in  51  =  (a,  5),  a^h. 

Let 

u  =  <^(x)  (1 

he  a  univariant  function  in  51  having  a  continuous  derivative  0'(a;)^O. 
Let  ^  =  (a,  yS)  he  the  image  of  21  afforded  hy  1).     Let 

X  =  yfr(u) 


372  PROPER   INTEGRALS 

he  the  inverse  function  of  (f).      Tlieii,  if  fl^jr^u^lylr' (u^  is  integrable 
in  ^, 

^J(x)dx  =  £f\_-^(u)-\-^\u)du.  (2 

By  358,  the  correspondence  between  the  two  intervals  21,  ^  is 
uniform. 

Let  -£'(%!,  ^2,  •••)  be  a  division  of  ^,  of  norm  S. 
Let  Aa;^  in  51  correspond  to  Aw^  in  ^. 
By  the  Law  of  the  Mean, 

Aa;«  =  '\/r'(77^)Aw^,         7]^  lying  in  Au^. 

Let  ^«  in  21  correspond  to  rj^  in  ^.     Then 

Ifa.^Ax^  =  2/[^(^0]^'(^<)Aw..  (3 

are  limited,  and  integrable  by  hypothesis,  we  have  2)  by  passing 
to  the  limit  in  3). 

2.  If  the  conditions  of  1  are  not  satisfied  in  the  intervals  21,  ^,  it 
may  be  possible  to  divide  them  up  into  subintervals,  in  each  of 
which  these  conditions  hold. 

541.    1.  Let  us  evaluate 

We  set 

X  =  tan  u  =  i/'(m),    or  u  =  arc  tan  x  =  <p(x). 

Then 

5t=(0,  1),    33  =(0,  w/4). 

The  conditions  of  540  are  obviously  satisfied.     Hence 
J'=|7    log  (I  +  tan  u)du. 

Let  us  make  a  new  transformation 

u  =  7r/4  —  V. 
The  conditions  of  540  being  again  satisfied, 

J=  l/^^log  1 1  +  tan  /^  -  x,U  dv. 


CHANGE   OF   VARIABLE  373 


But 

Hence 

or 
Thus 


tan 


(P')4 


—  tanr 
+  tan  V 


>  1  +  tan  V  Jo 

2  J- =  log  2  C'^*dv=- log  2. 


J  =  7r/8  log  2.  (2 


2.  That  we  should  not  affect  a  change  of  variable  in  a  definite 
integral,  without  due  precaution,  is  illustrated  by  the  following 
example. 

Let  XVw^^^-£rf^=[-''*g]!..=i-  (3 


Let  us  change  the  variable,  setting 

x  =  -  =  ^(u). 
u 

Then 

a  =  -l,    6  =  1;     a  =  -l,    /3  =  L 


^\AKn)WWau  =  -  £  ^,  =  - 1-  (4 


The  two  integrals  3),  4)  are  not  equal.     The  reason  for  this  is  that  the  function 

u  =  (p(x)=  - 

X 

of  540  does  not  have  a  continuous  derivative  in  2l=(— 1,  1).     Indeed,  it  is  not 
even  defined  throughout  %. 

542.  Let  X  =  -v/r(w)  have  a  continuous  derivative  in  S&  =  («,  yS), 
«^/3.  Let  51  be  the  image  of  ^.  Let  f(x)  he  limited  and  inte- 
grahle  in  91,  and  let  f[^(u)^y^'(u)  he  integrahle  in  Sd-      Then 

jjix)dx  =£f\:s^(u)W(i'^^du.        a  =  ^(«),  h  =  ^(yS).      (1 

1.  Let  us  note  first  the  difference  between  this  theorem  and  that 
of  540.  In  540  i|r(w)  is  univariant,  and  91,  ^  are  in  uniform 
correspondence. 

In  the  present  theorem,  i/r  may  have  any  number  of  oscillations 
in  ^.  Furthermore,  the  intervals  21  and  (a,  5)  may  not  be  the 
same. 


374  PROPER   INTEGRALS 

Example.  Let  a;  =  >/'(?<)  =  sin  m,  S3=(0,  ^-v).  Then  the  image  of  S3  is  the 
interval  91  =  (  —  1,  1).  On  the  other  hand,  a  =  sin  0  =  0,  ft  =  sin  ^-  tt  =  J.  Thus 
(a,  6)  =  (0,  ^)  is  different  from  21. 

Let  /(x)  =  X.     Then 

/dx  =  i  "xdx  —  l;    \    f{ypu)\j/'udu=  \         sin  «  cos  « d«  =  }. 
Thus  the  two  integrals  are  equal,  as  the  theorem  requires. 
2.  To  prove  the  formula  1),  consider 

We  shall  show  that  F(u)  is  a  constant  in  ^.     As  it  is  0  at  a, 

^=0  throughout  ^. 

To  this  end  we  show  -r,, .  .      ^  .     n« 

F'(u)=  0,         m  ^. 

Then,  by  400,  2,  .^(w)  is  a  constant  in  ^,  and  therefore  0. 
At  any  point  u  of  -33,  we  have 

/(^)c^:r-J  ^(w)^w.  (2 

There  are  two  cases : 

1°.  yjr'C^u^  T^  0.  Then,  by  403,  ■\lr(u)  is  univariant  in  V(u). 
We  can  thus  apply  540.     Hence 

AF=0,         in  VCu'), 

and  therefore  tt/      n 

F'  =  0,         at  M. 

2°.  '\jr'(^u)=  0.  Let  us  apply  the  theorem  of  the  mean  532  to 
each  integral  in  2).     Then 

AF  =  ^Ayfr  -  ^Am, 

where  ^  =  Mean /(a:),   ^  =  Mean  ^(«*) 

in  Ayfr,  Au  respectively.     Thus 

Au  Au 


CHANGE   OF   VARIABLE  375 

Am      ^  ^  ^ 
lim  ^  =  0, 
since  /['^(w)]  is  limited  in  51,  and 

limi/r'(w)  =  ',^'(w)=0, 

tl=U 

-»/r'  bei9g  continuous.     Hence  F'  (u)=  0,  also  in  this  case. 

543.  ltetf(x)  he  limited  in  21  =  (a,  6),  a^5.  Xet  u  =  (f)(x^  have 
a  continuous  derivative  <^'  (x)  ^  0  in  21.  iei  ^  =  («,  yS)  5e  the  image 
of%;  a=z(fi(^a},  /8=^(6),  Jjet  x  =  -^(u)  be  the  inverse  function 
of  (f).     Then  _  _ 

jjdx=£f(if(u)^ir'(u)du,  (1 

j/dx  =  J[j^(i/r(w))  Vr'  (u-ydu.  (2 

Let  us  prove  1) ;  the  demonstration  of  2)  is  similar.  Since  <^ 
is  univariant,  the  intervals  21  and  ©  stand  in  uniform  correspond- 
ence by  358.     To  fix  the  ideas  let  ^  be  an  increasing  function. 

Then  by  881,  yjr' {u^  >0,  and  continuous.  Let  ^=  (wj,  u^,  •••) 
be  a  division  of  ®  of  norm  B  into  subintervals  Aw^,  to  which  cor- 
responds a  division  D=  (rc^,  x^,  •••)  of  21  of  norm  d  into  intervals 

Ax^.     Let 

L^  =  Max /(a;),  in  Aa:^. 

=  Max/(-v|r(M)),  in  Aw^. 

M^  =  Max/(i/r(w))-«/r'(w),  in  Au^. 

\  =  Min  i/r'  (m),         yi4^  =  Max  '«/r'(w),        in  Aw^. 

l'=Max|/|,  in  21. 

We  have  to  show  that 

S,,  =  S^Aa:,,         /S^  =  ^M^Au^ 

have  the  same  limits. 


376  PROPER   INTEGRALS 

Since  ^'(2;)  and  yjr'^u)  are  continuous,  they  are  uniformly  con- 
tinuous. Hence  d  and  8  converge  to  0  simultaneously.  For  this 
reason  for  any  8  <  some  Sg, 

fi,-\<.=^ [,         uniformly  in  «. 

By  the  Law  of  the  Mean, 

Ax^  =  -«/r'(vJAw„         v^  in  Aw,. 
Hence 

But,  obviously, 

Max/ Min  i/r' <  Max/i/r' ^  Max/ Max  t/t' ; 

if  Max/>0,  while  the  signs  are  reversed  if  it  is  <  0. 

Thus  in  either  case  M^  lies  between  L^\^  and  L^/x^.  Also, 
L^yfr'Cy^^  lies  between  these  same  bounds.     Hence 


/3-a\ 


Hence  ,^|^e|EAwJ  r. ^  r. 

I  p  —  « I 

544.  Let  X  =  ^(u)  have  a  continuous  derivative  in  SQ  =  (a,  /3), 
a  ^  /3.  Let  i/r'  vanish  over  a  discrete  aggregate  A,  hut  otherwise  let 
it  have  one  sign.  Let  %  =  (a,  J)  he  the  image  of  ^,  a  =  t/^(«), 
h  ==  i/^(/3).     J/  owe  0/  ^Ae  integrals 


eadsts,  the  other  does,  a7id  hoth  are  then  equal. 

To  fix  the  ideas  let  JJ  exist.  By  403  the  correspondence  between 
31,  ^  is  uniform. 

Let  us  effect  a  division,  of  norm  S,  of  ^.  Let  the  norm  of  the 
corresponding  division  of  31  be  77.  Let  ^^  be  those  intervals  con- 
taining no  points  of  A,  while  ^2  =  ^  —  ^1  i^  ^^®  complement  of  ^j. 

Let  3li,  3I2  correspond  to  ^j,  ^2  respectively. 


seco:nd  theorem  of  the  mean  377 

Now 


But,  by  543, 

while,  since  A  is  discrete, 

lim   f  =0. 

6=0   -/S^ 

Hence  1)  gives 

J=  lim    I     =  lim   I 
33        «=o   »^2t^        ,,=0  »/2lj 

=  X-     by  522,  3. 
A  similar  reasoning  holds,  if  we  assume  that  X  exists. 


Second  Theorem  of  the  Mean 

545.    Let  f(x)  he  limited  and  integrable  in  31  =  (a,  5). 
Let  gix)  he  limited  and  monotone  in  2t.     Then 

jjgdx  =  g{a  +  0)£fdx  +  g(h -  ^')£fdx,  a<^<h.        (1 

Since  g  is  limited  and  monotone,  it  is  integrable  in  21  by  502. 
Hence  fg  is  integrable,  by  505. 

By  277,  8,  ^^^  _^  ^^^  ^^^  _  ^^  ^^.^^^ 

If  ^(a+ 0)  =  ^(6  —  0),  the  relation  1)  is  obviously  true.     We 
therefore  assume  that  these  limits  are  different. 
To  fix  the  ideas,  let  g(x^  be  monotone  increasing. 
We  begin  by  effecting  a  division  of  21, 

I)(ia^,  ^2  ••■  ««-i)' 
of  norm  8.     We  also  set 

a  =  ^Q,         h  =  a^. 

^^*  i^,  =  Max/,         m,=  Min/, 

in  the  interval  ^       ^  \        ' 


378  PROPER  INTEGRALS 

Let  ^^  be  any  point  in  8^. 

'''''*™  MX>KL')K>rnA,  (2 

and  also  r 

From  2),  3)  we  have 

Hence  r 

fQjK=Xfdx+cr^,  (4 

Multiply  4)  by  ^(f«)  ;  and  letting  «;  =  1,  2,  •••  w,  let  us  sum  the 
n  resulting  equations.     We  get 

Now  /^        /»&  rf> 

or  more  briefly,  ^        ^ 

~  Jk-\      Jk 

Hence  letting  «  =  1,  2,  •  •  •  w,  we  get 
Adding,  we  have, 


SECOND   THEOREM  OF   THE  MEAN  379 

Since  g  is  monotone  increasing, 

Let  9)Z  be  the  maximum  of  the  integral  I  fdx^  and  m  its  minimum 
as  X  ranges  over  21.     Then 

and  adding, 

Thus  7)  gives 

X  [  ^(^0  -  ^(^«-i)  }X-/^^ = ®  { ^^^"^  -  ^^^1^ } '    ^^ 

'^^'®''®  m<©<91W.  (9 

Thus  5),  6),  8)  give 

In  this  equation  let  8  =  0.     The  limit  of  the  left  side  is 

fjgdx. 
^^'°        lim  gC^,)  =  g(a  +  0)  ;    lim  gQ,-)  =  g(h  -  0). 

^'*  IK^)l<r; 

^^^'^  |2er,^(|J|<r2<.,. 

^'^^  S^.^9/,     by  494. 

As  /  is  integrable,  Un^^j^O; 

hence  the  last  term  of  10)  has  the  limit  0. 

Thus  all  the  terms  of  10),  besides  that  in  ©,  have  finite  limits. 
Hence  the  limit  of  @  exists.     Call  it  6. 


380  PROPER  INTEGRALS 

But  F  being  a  continuous  function  of  x,  it  takes  on  the  value  6 
for  some  point  |  in  21  by  357. 
Hence 

e=  Cfdx. 

Passing  now  to  the  limit  in  10),  we  have 

jjgdx  =  gia  +  O^jjdx  +  |  ^(5  _  0)  -  g^a  +  0)  |  j^fdx.      (11 

But  since  r*_  r^      rf> 

%)a         «/a         c/f 

we  get  1)  at  once  from  11). 

546.  If  g{x)  is  not  monotone,  the  formula  1)  of  545  may  not 
be  true,  as  the  following  example  shows. 

Let 

f{x)  =  x'^,        g{%)  =  cos  X. 
Then 

\       x^cosxdx^O,      by  527,  L 
since  the  integrand  is  never  negative  and  is  in  general  positive.     On  the  other  hand, 
g(a  +  0)  =  cos  -  7r/2  =  0  ;         g(b-0)=  cos  ir/2  =  0. 

Hence  the  right  side  of  1),  545,  is  zero.     The  formula  1)  is  thus  untrue  in  this 
case. 

INDEFINITE    INTEGRALS 

Primitive  Functions 

547.  1.  The  theorem  of  538  is  of  great  importance  in  evaluating 
integrals.     For,  to  find  the  value  of 


=  rfdx,  (1 

%/  a 


f(x)  being  limited  and  integrable  in  21  =  (a,  x),  we  have  only  to 
seek  a  function  F(x)  which  is  one-valued  in  21  and  has  f(x)  as 
derivative.     Then,  as  we  saw, 

J=F{x)-F{a).  (2 


PRIMITIVE   FUNCTIONS  381 

Let  G-(x)  be  any  other  function  which  is  one-valued  in  51  and 
has /(a;)  as  derivative.     Then 

J=  G^x')  -  6^(a).  (3 

Comparing  2),  3),  we  have 

a<ix^  =  FQx')  +  (7, 
where  (7  is  a  constant. 

The  functions  F(jic)^  ^(.^^  ^^'^  called  primitive  functions  of  f(^x) . 
They  are  denoted  by 

jfCx^dx, 

no  limits  of  integration  appearing  in  the  symbol.  Primitive 
functions  are  also  called  indefinite  integrals.  In  contradistinction, 
integrals  of  the  type  1)  are  called  definite  ititegrals. 

2.    Every  formula  of  differentiation.,  as, 

J),F(x-)  =/(a:), 

/(a;)  being  limited  and  integrable  in  an  interval  21,  gives  rise  to  a 
formula  of  integration, 

jf<ix~)dx=F(x). 

For  reference,  we  add  a  short  table  of  indefinite  integrals.* 
We  observe,  once  for  all,  that  in  all  formulae  involving  indefinite 
integrals,  we  shall  suppose  that  the  integrands  are  one-valued, 
limited,  and  integrable  in  a  certain  interval  21,  while  the  functions 
outside  the  integral  sign  are  one-valued  in  21- 

548.  I  adx  =  ax. 

x'^dx  = ,         it  ^  —  1. 

/x-fl 

/^  =  iogH. 

I  e^dx  =  e^. 

*Aii  excellent  table  of  integrals  is  "A  Short  Table  of  Integrals"  by  B.  O.  Peirce. 
Ginn  &  Co.    Boston. 


882  PROPER  INTEGRALS 

Ja^'dx  =  ,  ^     ,         «>0. 
log  a 

/dx  1.x  n 

a'^  +  x^      a  a 

/dx  .    X 

=  arc  sin  -,  a^O. 

Va^  —  x^  * 

I  sin  xdx  =  —  cos  a;. 
I  cos  xdx=  sin  a:. 
I  tan  xdx=  —  log  [  cos  a;|. 
I  cot  xdx=  log  1  sin  x\. 
I  tan  a;  sec  xdx  =  sec  a;. 
I  sec^  xdx  =  tan  x. 

549.  Not  every  limited  integrable  function  in  51  =  (a,  6)  has  a 
primitive,  as  we  now  show. 

Let /(a;)  be  continuous  in  21;  let 

F(x)=  I  fdx,         a<x<b. 

Then,  by53T,  ,^ 

^=/(a.),  in  21. 

dx 

Let  us  define  a  limited  function  gQjc)  in  %  as  follows :  it  shall 
=f(x)  except  at  points  of  a  discrete  aggregate  in  A,  at  which 
points /(a;)  :^^ (a;).     Then,  by  530, 

Cgdx  =  F(x),  (1 

Suppose  now  gCx)  had  a  primitive  G(x)  in  2t.     Then,  by  538, 

j^gdx  =  a(x)  -  a(a).  (2 


METHODS   OF   INTEGRATION  383 

Comparing  1),  2),  we  get 

Gr(x)  =  F(x)  +  (7,         C  a  constant. 


Hence  ,^       ,,-, 

dG     dF 


in  21; 


dx       dx 

or  f(x)  =  g(x),  in  21, 

which  is  a  contradiction. 

Methods  of  Integration 

550.  In  order  to  find  the  primitive  of  a  function  f(x)  we  may 
proceed  as  follows  :  We  first  consult  a  table  of  indefinite  integrals. 
If  the  integral  we  are  seeking  is  not  there,  we  try  to  transform  it 
into  one  or  more  integrals  which  are  in  the  table. 

The  principal  transformations  employed  are  : 

1°.   Decomposition  of  the  integrand  into  a  sum. 
2°.  Integration  by  parts. 
3°.  Change  of  variable. 

We  treat  these  now  separately. 

551.  Decomposition  of  the  integrand  into  a  sum.  This  method, 
as  its  name  implies,  consists  in  breaking  f(x')  up  into  a  sum  of 
simpler  functions.     Thus,  if 

f(x)=f^ix)+"-+fs{x\ 

then  lfdx=  {f-^dx  +  •••  +  )fsdx» 

J  =  \  cos2  X  dx. 


Ex.  1. 


As 


COS''  X  = 


2 

J"  =  H  (?x  +  M  cos  2xdx 
=  I «  +  ^  sin  2  X. 


384  PROPER  INTEGRALS 

Ex.  2. 


J  a 


+  ^^dx. 


+  i3x 


Since 


Now 


Hence 


Thus 


a  +  bx^b_^   a^-ah  ^        by  91,  2), 

(fee  !(:?•(«  +  Sx)      1,1      /^  ,   o  N 

= ^^ —   '^  ^  =  -  «  •  log  Cce  +  fix). 

a  +  /3x/3      a  +  /3x         )3         *^        '^^ 


J  oc  4-  fix     fi 


^x     ^ 

/3  /32 


Integration  hy  Parts 

552.  In  the  interval  21,  let  w(a;),  v(a;)  be  one-valued  functions 
having  limited  integrable  derivatives.     Then 

DjUV  =  uv'  +  vu' . 

Hence  /»  ^ 

I  uv'  dx  =  uv  —  \  vu'  dx.  (1 

The  application  of  1)  to  e value 

is  as  follows.     We  write 

f=  uv' . 

Then  1)  shows  that  C    i  - 

J=  uv  —  \  vu'  dx. 

The  determination  of  J  is  thus  made  to  depend  upon 

j  vu'  dx. 

553.  ExoL  .     r  , 

t/  =  I  X  log  X  (^X. 

Set 

«  =  log  X,         «'  =  X. 


set 


Then 
Hence 

554.  Ex.2. 

Set 
Then 

Hence 

To  find 

t 
Then 

Hence 


CHANGE   OF   VARIABLE 

u'=-,        v  =  hx^. 

X 

(/  =  i  x2  log  X  —  I  ixdx 
=  |(logx-i). 

J=  i  e'^ sin  bx dx,        a,  b  =^0. 
u  =  sin  bx,        v'  =  e"^. 


u'  =  b  cos  bx,        V  =  -  e^' 
a 


385 


j  —  ^gax  sin  bx 
a 


a  J 
A"  =  I  e"*  cos  bx  dx, 

u  =  cos  bx,        v'  =  e" 


e«^  cos  bx  dx. 


u'  =—  b  sin  bx,         v  —  -  e"". 
a 

K=-  e"^  cos  6a;  +  -  I  e"^'  sin  bx  dx 
a  a  J 


e"^  cos  bx  +  -  J. 


J  _  e'"'{a  sin  bx  —  b  cos  bx] 
a^  +  fe2 


This  placed  in  1)  gives 
The  same  method  gives 

f  e"-  cos  bxdx=  ^"^^  ^°"  ^^  +  ^  "'"  H 


(1 


(2 
(3 


Change  of  Variable 

555.  Letf(x)  he  continuous  in  the  interval  %.  Let  u  =  <^(a;)  have 
a  continuous  derivative  ^'  (a;)  =^  0  m  21.  Let  •©  5e  ^Ae  image  of  31, 
a/ifZ  x  =  ylr(u^  be  the  inverse  function  of  (f).      Then  if 

J  /['^(**)]'^'  (u)du  =  G-(u),         valid  in  ^  ; 

I  /(a;) dx  =  G-  [^ (a;)  ] ,         valid  in  21  • 


For, 


dG-[^^(x)~\  _  dG(u')     du 
dx  du        dx 


0 


386  PROPER  INTEGRALS 

But  by  hypothesis, 

du     dx  _-. 
dx     du 


1)  gives 


556.  Ex.  1. 

Set 
Then 


Set 
Then 


558.  Ex.3. 

Set 
Then 


™^=/[^(„)]  =/(.). 


k 


dx 


xlogx 
u  =  loer  X. 


t7=  i  —  =  log  M  =  log  log  X. 
J  u 


557.   Ex.  2.  j^  C dx_ 


xV2  ax  —  a^ 


u  =  -\/2  ax  —  a^. 

=  21  — =  -  arc  tg  - 

J  a^+u^     a        ^  a 


a  a 


=1 


(?« 


Vx2  ±  a2 


J=  \  —  =  log  u 
J  u 

=  log(x  +  Vx2±a2). 


559.  Ex.  4. 

Set 
Then 
dn  integral  evaluated  in  Ex.  3. 


=1: 


dx 


V(x  —  a)(x  —  6) 
u  =  Vx  —  a. 


J=2 


J 


(Z« 


Vm2  +  a  -  6 


DEFINITIONS 


387 


INTEGRALS  DEPENDING  ON  A  PARAMETER 

560.    Let  the  rectangle  bounded  by  the  lines  x  =  a^  x-=h^  y  =  a^ 
^  =  /3  be  denoted  by 

I{=  (a,  5,  a,  yS). 

Let  21=  (a,  5),  «=(«,  y8). 

Let /(a;,  y^  be  defined  over  R.  If  we  give  to  y  an  arbitrary  but 
fixed  value  in  ^,  /"(a;,  ?/)  is  a  function 
of  X  defined  over  21.  But  since  its 
value  also  depends  on  the  particular 
value  assigned  to  y,  we  say/  is  a  func- 
tion of  X  which  depends  upon  the  param- 
eter y.  If  for  each  value  of  y  in  ^, 
f(x^  y')  is  a  limited  integrable  function 
of  X  in  2t,  we  shall  say  it  is  regular  in 
M.     When /(a;,  y')  is  regular  in  M, 


J(.y')= jyQ^y')^^ 


R 


(1 


defines  a  one-valued  function  of  y,  over  the  interval  ^. 

In  performing  the  integration  indicated  in  1),  we  consider  y  as 
constant  and  integrate  with  respect  to  x. 

In  the  present  section  we  propose  to  study  the  function  «/iCy) 
with  respect  to  contmuity,  differentiation,  and  integration. 


Ex.  1. 

21  =  (0,  tt),  S3  any  interval. 
Ex.2. 


'^(y^=Sr- 


J(y)  =  f'^log  (1  -  2  2/  cos  X  -I-  y^)clx. 


3(  =  (0,  tt),  S3  any  interval  not  including  y  =±1. 
For,  log  M  is  continuous  when  m  >  0.     Here 

M  =  1  —  2  2/  cos  X  +  y2  =  (?/  —  cos  x)2  +  sin^x  5. 0, 

and  hence  m  =  0  only  for  the  points  whose  coordinates  are 

X  =  W47r,  y  =  (- I)"*. 


(2 


388 


PROPER  INTEGRALS 


Continuity 

561.    Let  7]  be  an  arbitrary  but  fixed  value  of  ^z  in  ^  =  (a,  /3). 

Let  us  denote  the  line  y  =  ^  by  ^. 

Let  4^{x)  be  defined  over  21=  (a,  6). 

If  for  each  e  >  0,  there  exists  a  S  >  0,  such  that 

\f(x,'n^K)-^(x)\<e,        (1 

for    each    0<|A|<S,    and    every    x 
in    51,    we    say :    f(x,    y)    converges     ^. 
uniformly  to  <f>(^x)  along  the  line  H, 
or  with  respect  to  the  line  H. 
We  denote  this  by 


p 

a 

1 

a 

b 

or 


lim/(a;,  y')  =  (^(a;), 

y=ri 

f(x,  2/)  =  j>(x). 


uniformly  ; 
uniformly  along  H. 


If  in  the  relation  1),  only  positive  values  of  h  are  considered,  we 
say /(a;,  y')  converges  on  the  right  uniformly,  etc. 

If  only  negative  values  of  h  are  considered, /(a;^/)  converges  on 
the  left  uniformly,  etc. 

\if(xy^  converges  uniformly  to /(a;,  rf)  with  respect  to  the  line 
_ff,  we  shall  say  f(xy)  is  a  uniformly  continuous  function  of  y  with 
respect  to  the  line  H,  or  along  the  line  H. 

lif(xy')  is  a  uniformly  continuous  function  of  y  with  respect  to 
each  line  ^  =  ?;  in  ^  =  (a^S),  we  shall  ssij  f(^xy)  is  a  uniformly  con- 
tinuous function  of  y  in  ^. 

562.    1.  Letf(x,  «/)  be  regular  in  any  R  =  (a57/3),  a<7</3. 


Let 


It\vmf{x,  y}=(f>Qx) 


uniformly  along  the  line  y  =  cc. 

Let  4>(x)  he  limited  and  integrahle  in  21. 

1  hen  ^j  -J  ^j 

i^lim  I  f(xy~)dx=  I  R\\mf{xy^dx=  )  <^(x)dx. 

For,  let 
A  =  j/ix,  a  +  h~)dx  -  jj>(x^dx  =  Jjfix,  a  +  A) -  (^Qx')\dx.      (2 


(1 


CONTINUITY        -  389 

We  have  to  show  that 

e>0,  8>0,  lA|<e,  ()<h<8.  (3 

But  by  hypothesis,  for  each  e  >  0,  there  exists  a  S  >  0,  such  that 

\f(ix,a  +  h)-cf>(x-)\<-L-  (4 

o  —  a 

for  each  0<h<8,  and  any  2:  in  51. 

Hence  3)  follows  from  2),  4),  and  524,  2). 

2.  That  the  relation  1)  may  not  hold  when  /(a;,  ?/)  does  not 
converge  uniformly  to  ^(a;),  is  shown  by  the  following  example  : 


Let 

Here 

Hence 

On  the  other  hand, 

Hence 


/(a;,  y)  =   „  y    .,        for  X  =?!=  0  ; 
x2  +  2/2 

=  0,  for  X  =  0. 


i?lim/(x,  ?/)  =  0(x)  =  0. 

y=0 


j'V(x)(?x  =  0.  (5 

J=  \  fdx—  \     „  ^     dx  =  atctg~,        2/>0. 
Jo-'  Joa;2  +  w2  °« 


i?limj'=-.  (6 

y=0  2 


As  5),  6)  have  different  values,  the  relation  1)  does  not  hold  here.     Obviously 
/(x,  y)  does  not  converge  uniformly  to  0  in  any  interval  containing  the  origin. 

563.    1.  As  corollaries  of  562  we  have : 

Let  f(x,  if)  he  regular  in  R  =  (aba^~).     Let  it  he  uniformly  con- 
tinuous in  y,  along  the  line  y  =  r).     Then 

is  a  continuous  function  of  y  at  7].         ^^V^^- 

2.  Let  f(x,  3/)  he  regular  in  R  =  (^aha/B}.     Let  it  he  a  uniformly 
continuous  function  of  y  in  ^.      Then  JQy^  is  continuous  in  Sd- 


390  PROPEli   INTEGRALS 

3.  If  f(x,  ^)  be  continuous  in  B,Qaba/3},  JQy^  is  continuous  in 
«  =  («,y9). 

This  follows  at  once  from  352. 

Example.    In  538  we  proved  the  relation 

^^  -^     Jo      1  +  X2tan2x      2    1  +  |X| 

for  all  values  of  X.  It  required,  however,  a  separate  integration  to  establish  it  for 
X  =  ±  1.  By  the  aid  of  3,  we  may  prove  the  correctness  of  1)  for  these  values 
without  any  calculation.  Consider,  to  fix  the  ideas,  X  =  1.  Since  the  integrand  of 
1)  is  obviously  a  continuous  function  of  x,  X  in  the  band  B  =  (0,  ir/2,  1  —  S,  1  +  5), 
the  integral  is  a  continuous  function  of  X  at  1.     Hence  1)  holds  for  X  =  1. 

564.  The  results  of  562,  563  may  be  generalized  as  follows : 

Let  A  be  a  discrete  point  aggregate  in  51.  We  can  divide  51  into 
two  systems  of  intervals,  (5  and  3D,  such  that  (S  contains  no  point 
of  A,  and  the  total  length  d  of  the  intervals  T)  is  as  small  as  we 
please. 

We  shall  say/(aj,  y)  converges  uniformly  to  (j)Qx}  along  the  line 
«/  =  77,  except  at  the  points  A,  when,  for  each  e  >  0  and  any  (5,  there 
exists  a  S  >  0,  such  that 

1/(2;,  77  4-A)-<^(2;)|<6 

for  each  0  <  [  A|  <  S  and  every  x  in  (5. 

The  terms,  /(a;,  ?/)  converges  on  the  right,  or  on  the  left  uniformly, 
except  for  the  points  A,  need  no  special  explanation. 

Also  the  meaning  of  the  term  /(a;,  3/)  is  uniformly  continuous 
along  the  line  y  —  t],  except  for  the  points  A,  is  obvious. 

565.  Let  f(x,  y')  he  regular  in  the  rectangle  R(a,  b,  a,  yS).  Let 
f  converge  uniformly  to  <^(x)  along  the  line  y  —  'r],  except  for  the 
points  of  a  discrete  aggregate  A.  Let  <^{x)  be  limited  and  integrable 
in  51  =  («,  J).      Then 


f{x,y')dx=\    \\XQ.f{x,y^dx=  \  <^(x)dx,  a<'^<^« 

fix,  7}  +  h^dx  —  \ji{x^dx. 


Let 


CONTINUITY  391 

We  must  show  that 

6>0,        S>0,        \I>\<€,        0<\h\<S.  (1 

Since /is  limited  in  R,  and  <f>  in  21, 

\<l>(x')\,         \f(x,^)\<M,         inR. 

Choosing  e  >  0  small  at  pleasure,  and  then  fixing  it,  we  choose 
the  system  ®  such  that  its  length 

d<    ' 


4.M 

Then 


D  =  (\Kx,  r)  +  h:^-<f>(x)\dx+  flfCx,  ^  +  A) -  «^(a:) I dx. 
ence  ,  ^ ,      ,  ^  , 


But 


'©1*^2' 


IX 

On  the  other  hand,  by  hypothesis, 
for  each  0  <  |  A  |  <  S,  and  every  x  in  (5.     Hence 

IXNl 

Hence  2)  gives 

which  proves  1). 

566.    As  corollary  we  have : 

Let  f(x^  y)  he  regular  in  the  rectangle  R=  (a,  5,  a,  yS).  Let  it 
be  uniformly  eoiitinuous  in  y  along  the  line  y  =  ri.,  except  for  the 
points  of  a  discrete  aggregate  A.     Then 

is  continuous  at  ij^  «  <  ^  <  /3- 


392  PROPER   INTEGRALS 

Differentiation 

567.    1.   1°.  Letf(x,y')^f'y(x,y)heregulm'inR  =  (aha^'). 
2°.   Letf'y  he  uniformly  continuous  in  y  along  the  line  y  =  r],  ex- 
cept for  the  points  of  a  discrete  aggregate. 

Let  pt, 

Then  c^ 

J' On)  =j/yi^^  Vy^,  a<rj<0.  (1 

For, 

At^  A  Ja  h 

But  by  the  Law  of  the  Mean, 

f(x,  r]  +  h)-f(x,  rf)  ^  ^,         ^^^ 

where  f  lies  between  rj  and  rj  +  h  and  depends  on  x  and  h.     But 

by  2°, 

lim  o-  =  0,         uniformly  except  for  A. 
-Lhus  ^j-      ^^  ^j 

which  gives  1),  on  passing  to  the  limit,  A  =  0. 

2.  As  corollary  we  have  : 

Letf{x,  y^,f'y(x,  y')  he  continuous  in  the  rectangle  (ahajS').     Then 

,,        ^Cfix,yyix=Cfy(^x,y^dx.         a<y<^.      (3 

3.  Criticism.  Many  text-books  give  the  following  incorrect 
demonstration  of  1. .  From  2)  we  have,  changing  slightly  the 
notation, 

dJ  y  AJ  y  C'Af  f. 

—  =  lim  —  =  lim  I    -^  •  (4 

dy  Ay  •^«  Ay  '. 


DIFFERENTIATION  393 

It  is  now  assumed,  without  further  restriction,  that 

'A/ 

As 


*/a  Ay     ^« 


hm-r^- 


(6 


4)  and  5)  give 


i^y      dy 
ay     ^«  dy 


But  we  have  already  seen  in  562,  2,  that  an  interchange  of  the 
symbols 

lim,        f 

is  not  always  permissible. 


4.  Example. 


Here,  f{x,  y)  and 


J=  r'^log(l  —  2  y  cos  X  +  y^)dz. 


(6 


fy{^y) 


2  (y  —  cos  x) 
1  —  2  ?/  cos  X  +  ?/2 


are  continuous  in  the  rectangle  (0,  tt,  «,  /3)  if  (a,  /3)  does  not  contain  the  points 
y  =  ±l.     Cf.  560,  Ex.  2,     Hence 


5^  =  2^^ 
dw        Jo  1  - 


TT     (y  — cosx)dx 


2  y  cos  X  +  y2 ' 


12/1^1. 


(7 


5.  The  following  is  an  example  where  the  conditions  of  theorem 
1  are  not  satisfied,  and  where  differentiation  under  the  integral 
sign  leads  to  a  wrong  result. 


Let 


Then 


Hence 


Fix,  y) 


xy 


Vx'-^  +  y'^ 
=  0, 


D^F(x,  y)  = 

(x2  +  y2)t 

=  0, 
CD,Fixy)dx=     J__ 


except  at  origin  ; 
at  origin, 
except  at  origin  ; 

at  origin. 
y  arbitrary. 


394  PROPER  INTEGRALS 

Let 


/(x,  y)  — -,        except  at  origin ; 

(x2  + 2/2)2 

=  1,  at  origin. 


Then  ^,  ^.  y 


J<iy)=^/dx  =  j^'D.Fdx: 


Vl+J/2 

Hence 

J-'(0)=1.  (8 

On  the  other  hand, 

fy(x,  0)  =  0,  X  arbitrary. 

Hence  ^i 

^j;(x,0)dx  =  0.  (9 

The  equations  8),  9)  show  that  in  this  case  differentiation  under  the  integral 
sign  is  not  permissible.    In  fact,  we  observe  here  that 

3  x^y^ 
fyC^y  y)  = — 5^        except  at  origin  ; 

(X2  + 2/2)2 

=  0,  at  origin, 

and  is  therefore  not  limited  about  the  origin.     Thus,  condition  1°  of  theorem  1  is 
not  fulfilled. 

Integration 

568.  1.  Let/(aj«/)  be  continuous  in  the  rectangle  i2=  (a,  5,  a, /3). 
Since 

fdx  a<x<.h.  (1 

is  a  continuous  function  of  y  by  563,  2,  it  is  integrable  in  (a,  /3). 
Therefore 

is  convergent  for  each  «<  ?/</8. 

The  integral  2)  is  obtained  from  1)  by  integrating  with  respect 
to  the  parameter  y.  It  is  called  a  double  iterated  integral;  or  more 
shortly,  when  no  ambiguity  can  arise,  a  double  integral. 

2.  Let  /(a;,  y')  be  continuous  in  R  =  (jiba^).     Let 

dyj^fdx,         X,  y  in  R. 
^^'''  F'J,=  F'J^  =  f(xy),        inR.  (3 


INTEGRATION  395 

For,  by  537,  F',  =  £fd^ 

On  the  other  hand,  by  567,  2, 

=^£fdy,     by  537. 
Hence,  by  537,  F'J„  =  f(.y).  (5 

The  relation  3)  follows  now  from  4),  5). 

Inversion  of  the  Order  of  Integration 

569.  The  integral  of  568,  viz. 

£dy£fdx,  (1 

is  obtained  by  integrating  first  with  respect  to  a;,  and  then  with 
respect  to  y. 

But  we  might  have  integrated  in  the  inverse  order,  getting 

Sj.£fdy.  (2 

It  frequently  happens  that  the  integrals  1),  2)  are  equal,  and 
this  fact  is  of  greatest  importance  in  transforming  such  integrals. 
When  these  two  integrals  are  the  same,  we  say  that  we  can  invert 
the  order  of  integration,  or  the  integral  2)  admits  inversion.  We 
give  now  a  simple  case  when  this  inversion  is  possible.  Later  we 
shall  give  a  broader  criterion. 

570.  1.  Let  f(xy^  he  limited  in  the  rectangle  i2  =  (aJa/3).  Let 
it  he  continuous  in  R,  except  possibly  along  a  finite  numher  of  lines 
parallel  to  the  x  or  y  axes,  along  which,  however,  f  is  integrahle. 

X  ^yjj^^y^^^  ^  ja  dx^Jixy^dy.  (1 


396 


PROPER   INTEGRALS 


Case  1.     Let/  be  continuous  in  R  without  exception. 
We  saw  in  568  that 


Hence,  by  538, 


»x      Sy      -^  ^' 


dy 


For  the  same  reason, 


Similarly, 


dy         "      ^<-         By 
=  F(h,  y8)  -  F(h,  «)  -  FQa,  /3)  +  ^(a,  «) .       (2 


dx 


dx 


^^^  J^^^a:  J^%?/  =  jP(6y3)  -  FQajS)  -F(ha)  +  Fiaa) .  (3 

From  2),  3)  we  get  1). 

Case  2.     Let/  be  continuous  in  M^  except  for  points  on  the  line 

x  =  b. 

By  1  we  have,  a  <b'  <b, 

J^jjdy^jjyjjdx.        (4 
Moreover,  by  563,  3, 

being  a  continuous  function  of  x  in  (a,  5}  except  possibly  at  b,  and 
limited  in  (a,  6), 

fjxffdy 

exists. 

Now,  by  536, 

lim   I   c?2;  I  fdy  =  \  dx  )  fdy.  (5 


INVERSION   OF   THE   ORDER  OF   INTEGRATION  397 

On  the  other  hand, 

Jdx  =  jJdx-J^Jdx, 

But  since  /  is  limited,  let 


|/l<Jf,         ini2. 


Then 


J-      -6 


<M(b-b'}. 


Hence  the  right  side  of  4)  gives 


Thus 


I  Cc^xCfdy-  Cdy  Cfdx\<M(^  -  a){h  -h'}. 


Letting  b'  =  b  and  using  5),  we  get  1). 

Evidently  the  same  reasoning  applies  Avhen  /  is  continuous  in 
M,  except  on  one  of  the  sides  of  M  parallel  to  a;-axis. 

Case  3.     Let  /  be  continuous  in  M  except  on  the  two  lines, 
x  =  b,  y  =  ^. 

Let  a<l3' <I3.     Then,  by  Case  2, 


S'dy£fdx=£dx£fdy. 

We  can  reason  with  this  equation  in  the 
same  manner  as  v/e  did  with  4).  This 
proves  the  theorem  also  for  this  case. 

Case  4.  lif(xy')  is  discontinuous  within 
i?,  we  have  only  to  divide  R  into  four 
rectangles  9?,  as  in  the  figure. 

Then  each  of   the   rectangles  ^  falls  - 
under  Case  3.     By  breaking  up  the  given 
integrals  into  these  rectangles  9?,  we  prove  1)  readily. 


398  PROPER  INTEGRALS 

Case  5.  Creneral  Case.  By  breaking  R  into  smaller  rectangles 
di,  bounded  by  the  lines  on  which  the  points  of  discontinuity  lie, 
we  reduce  this  case  to  Case  4. 

2.  We  have  shown  in  1  that  inversion  is  permissible  when/(a;y) 
is  continuous,  except  at  points  lying  on  a  finite  number  of  lines 
parallel  to  the  x  and  i/  axes.  It  will  be  shown  in  Chapter  XVI 
that  inversion  is  permissible  under  much  wider  circumstances. 

The  points  of  discontinuity  may  lie  on  an  infinite  number  of 
lines ;  moreover,  these  lines  do  not  need  to  be  parallel  to  the  axes ; 
they  do  not  even  need  to  be  right  lines. 

Example.     We  saw  by  538,  2,  that 

dX  TT  1 


Jo       1 


+  ?/2  tan2  X     2     I  +  \y\ 
Hence 


J=  Cdy  r" '^ =  ^  r-^L.^Elog2.  (1 

Jo    ^  Jo      1  +  ^2  tan2  x2  Jo   1  +  2/2  ^ 

As  the  integrand  is  limited  in  the  rectangle  ( 0  -  01 J ,  and  is  discontinuous  only 
when  X  =  ir/2,  we  can  invert  the  order  of  integration.     Hence 

J=  r^'dx  C '^y  =  C^^'a^[^rctg(yt^nx)l^ 

Jo  Jo  1  +  y2  tan2  a;     Jo  L        tan  x        Jo 

_    r^r/Z  xdX  _  xo 

Jo       tan  X 

C'/'xd^^ZiogH. 
Jo      tansc     2 


Thus  1),  2)  give 


TO 

^  q 

en  o 
Q 


o 


r 


CHAPTER   XIV 
IMPROPER  INTEGRALS.    INTEGRAND  INFINITE 

Preliminary  Definitions 
571.    1.  Up  to  the  present,  we  have  considered  only  integrals 


jj(x)dx, 


in  which  the  integrand  is  limited,  as  well  as  the  interval  of  inte- 
gration 21  =  (a,  5). 

It  is  desirable  to  extend  the  definition  of  an  integral  to  embrace 
integrands  and  intervals  of  integration  which  are  not  limited. 
Such  integrals,  we  said,  are  called  improper  integrals. 

In  this  chapter  we  consider  improper  integrals  for  which  21  is 
limited,  and /(a;)  is  unlimited  in  21- 


dx 


Vl-a;2 

are  examples  of  such  integrals. 

2.  The  reader  will  observe  that  the  integrand  of  J  is  not  defined 
at  a;  =  0,  and  the  integrand  of  ^  at  a:  =  1.  When  convenient,  we 
may  assign  to  the  integrand  at  such  points  any  value  at  pleasure. 
Cf.  598. 

572.  Let /(a;)  have  a  finite  number  of  points  of  infinite  discon- 
tinuity in  21,  347, 

C\i    ^2'    '"    ^m' 

We  shall  call  these  singular  points^  and  say  f(x)  is  in  general 
limited  in  21 ;  or  that  it  is  limited  except  at  these  points.  Let  us 
inclose  each  point  c^  within  a  little  interval  S^  containing  no  other 

399 


400  IMPROPER  INTEGRALS.     INTEGRAND   INFINITE 

singular  points.  Let  ^  be  what  is  left  after  removing  the  inter- 
vals (5k  from  21.  On  varying  the  intervals  S^,  ^  will  vary.  If  for 
each  choice  of  ^,  f(x)  is  integrable  in  ^,  we  say  f(jc)  is  regular 
in  %  except  at  the  points  Cj  •  •  •  <?^ ;  or  we  say  f(x)  is  in  general 
regular  in  21. 

573.    1.   Let /(a;)  be  regular  in  2l  =  («,  5)  except  at  h.     If 

lim  I  f(x)dx,         a<8<h,  (1 

is  finite ;  we  say  f{x')  is  integrable  in  31,  and  define  the  symbol 

J=jJ(oc)dx 

to  be  this  limit. 

Similarly,  \if(x)  is  regular  in  21  except  at  a,  and 

lim  I  f(x)dx^  a<a<h^  (2 

is  finite;  we  say  f(x)  is  integrable  in  2t,  and  define  J  to  be  this 
limit. 

Finally,  if /(a:)  is  regular  in  21,  except  at  a,  b,  and 

lim  I  f(x)dx,         a<a</3<b,  (3 

a=a  "^  o- 

is  finite ;  we  say  fix)  is  integrable  in  21,  and  define  J  to  be  this 
limit. 

When  these  limits  1),  2),  3)  are  finite,  we  say  J"  is  jinite  or  con- 
vergent. When  these  limits  are  infinite,  we  say  J  is  infinite.  If 
these  limits  do  not  exist,  finite  or  infinite,  we  say  J  does  not  exist. 

we  have  taken  a<h.  Precisely  similar  definitions  would  apply  if 
a>b. 

Obviously, 

f/dx=-jjdx.  (4 


PRELIMINARY   DEFINITIONS  401 

For,  to  fix  the  ideas,  suppose  fix)  regular  except  at  «,  and  let 
the  integral  on  the  right  be  convergent.     Then,  if  a<a<h, 

jjdx=-£fdx,  by  524,  1). 

Passing  to  the  limit,  we  get  4). 

574.    1.  Let  /(^)  be  regular  in  %  =  (a,  5),  except  at  h. 
For  fix)  to  be  integrable  in  31,  according  to  the  definition  just 
given,  it  is  necessary  and  sufficient  that 

6>0,  g>0,  I  (^dx-  Cfdx\<e 

for  any  pair  of  numbers  /3,  ^'  within  (h  —  S,  5),  by  284. 

Ja  Ja  Jfi 

it  is  necessary  and  sufficient  that 

\^^fix^dx\<t,  (1 

The  integral  ^a- 

\  fix)dx,         5-8</S,  yS'<5,         (2 

is  called  the  left-hand  singular  integral  of  norm  Bfor  the  point  b. 
Similarly, 

I  fdx,  a<a,  a'  <a  +  8, 

is  called  the  right-hand  singular  integral  of  norm  8  for  the  point  a. 
We  have  thus  this  result : 

Let  f(x)  he  regular  m  51  =  (a,  5)  except  at  an  end  pointy  say  h. 
For  f{x)  to  he  integrable  in  31,  it  is  necessary  and  sufficient  that  the 
singular  integral  at  b  have  the  limit  0  ;   i.e. 

lim  Cf(x)dx  =0,  a<b-8</3,  13'  <b. 

2.  When  a  singular  integral  such  as  2)  converges  to  zero  as  its 
u  orm  8=0,  we  shall  say  the  singular  integral  is  evanescent. 


402  lAlPROPEli    INTEGRALS.     INTEGRAND   INFINITE 

3.  Let  f(x)  he  regular  in  51  =  (a,  5)  except  at  the  end  points  a,  h. 
For 

to  he  convergent^  it  is  necessary  and   sufficient   that  the  singular 
integrals 

Xa"  /»6" 

fdx,        Jj./*^^'         a<a' <a'\         h' <h"  <h 

he  evanescent. 

It  is  necessary.     For,  when  J  is  convergent, 

e>0,         S>0,         L/-  P|<1,        a<a<a  +  ^,         b-B<l3<h. 

<-,         a<a'  <a  +  B. 

"■'  I     2 

Adding, 

I  ,    <  e,  a<a^  a'  <a-\-S. 


Thus  the  singular  integral  at  a  is  evanescent.  The  same  is  true 
for  h. 

It  is  sufficient.  For  the  singular  integrals  being  evanescent, 
we  have 

J  "a"  I  I     /•ft"  I 

a      I        2  l*^*      I        2 

Hence  ,     ,,        ,„,  .. 

Hence  ^^ 

lim     I  /c?2:,          a<a<l3<h 

o=o,  ^=1^°- 

is  finite,  and  hence  by  definition  t7  is  convergent. 


575.    Ex.  1. 


Now,  forO</3<l,  ^„      ^^ 


0  Vl  -  X2 

i^Ccc)  =  P      ^^      =  arc  sin  /S. 


PRELIMINARY   DEFINITIONS  408 

As 

lim  arc  sin  |3  =  ir/2, 
J=ir/2.  P=i 

The  singular  integral  at  1  is 

ri3      dx  ... 

I     —=  =  arc  sin  /3'  —  arc  sin  |8.  (1 

•'^   vl  —  x^ 
As 

limarcsina;  =  7r/2, 

z=l 

the  difference  on  the  right  of  1)  is  numerically  <e  in  Vq(1),  for  a  sufficiently  small  S, 
Ex.2.  .,, 


Jo   x' 
Here,  for  0<a<l,  ^^ 

j   «5  =  _ioga. 


But 


J?  lim  loga  =  —  00. 

0=0 


Hence  J" is  infinite,  viz.  J=  +  cc. 
The  singular  integral  at  0  is 

f""^  =  log^,        0<a'<a"<5. 

Ja'     X  a' 


But 

does  not  exist,  by  321. 
Ex.  3. 


lim  log  ^=— 
s=o        a' 


Now,  ifO<a<l,  -1  ^ 


J-=rV-i(te,        X>0. 

I" 


But 


Hence 


lim?-li^  =  l,        by  299,  2. 
«=o      X  \ 


^=1. 


576.    Letfix)  he  integrahle  in  5t=  (a,  6),  awe?  regular  except  at 
a,  6.     igi  c  he  any  point  within  21.      Then 

fdx=jjdx+j/dx.  (1 

For,  since  J"  is  convergent, 

e>0,     S>0,     |j--ri<i,    a<a<a  +  h;     h-B<^<h.     (2 


404  IMPROPER   INTEGRALS.     INTEGRAND   INFINITE 

By  574,  3,  the  integrals 

fdx,  I  fdx 

are  convergent.     We  can  therefore  take  h  such  that  also 

lf_fl<j,   |/-r|<£. 

|»/0  ,Ja     I  4  W'^  ^<^      I  4 

Adding  these  last  ineqrialities  gives 

i(X'+x>r!<i-  cB 

Adding  2),  3)  gives 

From  this  follows  1).  - 

577.  1.  Definition.  We  can  now  generalize  as  follows.  Let 
f(x)  be  regular  in  51  except  at  the  singular  points 

a  <  Cj  <  (?2  <  •  •  •  <  c„j  ^  5. 

lif(x)  is  integrable  in 

(a,  Cj),    (cj,  ^2),    •••    ((?,„,  S), 

we  say /(a;)  is  integrable  in  31,  and  set 

/c?2;=  ]  fdx+i  fdx+-  +      /^a:. 

2.  We  can  therefore  say  : 

Letf{x)  he  regular  in  31=  (a,  5),  except  at  certain  points  Cj,  Cg,  •••  e^. 
For  f{x)  to  he  integrable  in  3t,  i^  *s  necessary  and  sufficient  that  the 
singular  integrals  at  c^,  c^.,  •••  he  evanescent. 

3.  From  this  follows  at  once 

Iff(x)  is  in  general  regular  and  is  integrable  in  31,  it  is  integrable 
in  any  partial  interval  of%. 


CRITERIA  FOR   CONVERGENCE  405 

4.  To  avoid  confusion  and  errors  of  reasoning,  the  reader  should 
remember  that,  when  f(x~)  is  not  limited  in  21  but  yet  integrahle 
in  21,  there  are  only  a  finite  number  of  singular  points  in  21 ;  and 
/  is  limited  and  integrable  in  any  partial  interval  of  21^  not  em- 
bracing one  of  the  singular  points. 

Critei'ia  for  Convergence 

578.    1.  The  integral  ^ 

K=fjfix}\dx 

is  called  the  adjoint  integral  of 

J—\  f(x)dx. 
We  write  K=AdjJ. 

Letf(x)  he  in  general  regular  in  21.     If  the  adjoint  of 

J=  Cfdx 
is  convergent^  J  is  convergent. 

Let  c  be  a  singular  point  of  /(a;).  We  wish  to  show  that  the 
singular  integrals  at  this  point  are  evanescent.  To  fix  the  ideas 
let  us  consider  the  left-hand  singular  integral.     By  528, 

fdx\<\    \f\dx,         c— 8<7<7'<c. 

By  hypothesis,  the  integral  on  the  right  vanishes  in  the  limit 
8  =  0.  Hence  the  integral  on  the  left,  which  is  the  left-hand 
singular  integral  at  e,  is  evanescent. 

When  AdjJ  is  convergent  in  21,  J  is  said  to  be  absolutely  conver- 
gent in  21  and /(a;)  is  absolutely  integrable  in  21. 

2.  Letf(x)  be  absolutely  integrable  in  21,  and  in  general.,  limited. 
Then  f  is  absolutely  integrable  in  any  partial  interval  of^. 

The  demonstration  is  obvious. 

3.  The  reader  should  note  that  /(a;)  may  be  integrable  in  21 
and  yet  not  absolutely  integrable,  as  the  following  example  shows. 


406  IMPROPER   INTEGRALS.     INTEGRAND   INFINITE 

Let  us  divide  the  interval  21  =  (0,  1)  into  partial  intervals,  by  inserting  the  points, 
1    1    1    ... 

5.  ?»?'  /      1  1\ 

In  the  interval  2l„  =  ( ,  -  ) ,  n  =  1,  2,  •••  let  us  erect  a  rectangle  jB„  of  area 

V«  +  i     "/ 
-.     Let  these  rectangles  lie  alternately  above  and  below  the  x-axis.    In  the  interval 
n 

2l„,  excluding  the  left-hand  end  point,  let  /(x)  =  height  of  Bn  taken  positively  or 
negatively  accordingly  as  Bn  is  above  or  below  the  axis.  Then  /(x)  has  a  singular 
point  at  X  =  0. 

We  have  now, 


Also, 


Cfdx  =  lim  Cfdx  =  lim  Cfdx  =  Hm  f  1  -  ?^  +  1 +  (^li)!!\ . 

JO  a=0  Ja  m=<Kj'i_  m=x  \  2        3  m  —  1   / 

C\f\dx  =  \im{l  +  l  +  l  +  ...  +  -J—). 
Jo  »i=«  \        2      6  m  —  lj 


As  the  reader  probably  knows,  or  as  will  be  shown  later,  the  first  limit  is  finite, 
the  second  infinite.    Thus  /  is  integrable  but  not  absolutely  integrable  in  9t. 

4.  The  reader  shoiild  note  this  difference  between  proper  and 
improper  integrals,  ^^ 

J=  j  f(x)dx. 

If  J  is  an  improper  integral,  we  have  just  seen  that  J"  is  conver- 
gent if  J 

K=\   \f(x^\dx 
is  convergent. 

But  if  t7  is  a  proper  integral,  we  saw  in  528,  2,  that  we  could 
not  conclude  the  existence  of  J  from  that  of  K. 

On  the  other  hand,  if  J"  is  a  proper  integral,  the  existence  of  K 
follows  from  that  of  J,  by  507 ;  while  if  J  is  an  improper  integral, 
we  cannot  conclude  the  convergence  of  K  from  that  of  J",  by  3. 

579.  1.  The  fi  test.  Letf(x')  be  regular  in  51  =  (a,  h')  except  at  a. 
For  some  0  <  /*  <  1,  and  M>0,  let  there  exist  a  F*(a)  sueh  that 

(ix-ay\f(ix)\<M,  in  V*. 

Then  f(x)  is  absolutely  integrable  in  21. 
Consider  the  singular  integral  at  a.     We  have 

-^»  -^  (x-ay      l-/ul  ^  ^  ^      J 

which  vanishes  in  the  limit,  l)v  299,  2. 


CRITERIA  FOR  CONVERGENCE  407 

2.  As  a  corollary  we  have: 

Let  fix)  he  regular  in  %  =  (a,  5)  except  at  a. 
For  some  0  <  yii  <  1,  let 

R\im(x-ay\f(x)\ 

he  finite.      Then  f(x)  is  absolutely/  integrahle  in  %. 
Ex.  1. 


is  convergent. 

For,  by  454,  Ex.  2, 


i  \ogxdx 


B  lim  x'^  I  log X I  =  0,        /u>0. 
Ex.  2. 


is  convergent. 
For, 


\  X  ^  sm-  dx 
Jo  X 

iJlima;'*  .  |x~^sin-|  =0,        M>f 


580.    Let  f(x)  he  regular  in  %  =  (a,  5)  except  at  a. 
In  V*{a)  let  f(x)  have  one  sign  a,  while 


Then 


ix-ayf(x)>M>0. 
J=  i  fdx  =  a- CO. 


For,  let  a<a<c<a  +  8. 


Then 


I  fdx\>  I  dx  =  M\og 

\Ja'^      \—Jo.  X  —  a  °a—a 

=  +  00,  when  a  =  a. 

I  fdx  =  0-  •  CO. 
Example.  /•!    ^^j. 


Hence 


Jo  1  _  a;2 


+  09. 


For, 
for  X  near  1. 


(i-.y(x)=^>l. 


408 


IMPROPER  INTEGRALS.     INTEGRAND   INFINITE 


581.    Let  f(x)  he  regular  in  %  =  (a,  5),  except  at  a. 
^^^  \=\im(x-a^f(ix) 

x=a 

exist.     If  f(x)  is  integrahle^  \  must  he  0. 

Let  us  prove  the  theorem  by  showing  that  the  contrary  leads  to 
a  contradiction. 

To  fix  the  ideas  suppose  \  >  0.     Then,  for  each  /i  such  that 

0</u,<A, 

there  exists  a  V^*(a^  such  that 

(x  -  a}f(pc)>  fi,         in  F/. 
Then  the  singular  integral, 


iJf>0,     yu>0, 


Jdx  =  +  co,     by  580. 

Hence  /  is  not  integrable  in  %. 

582.  The  criteria  of  579,  580  admit  a  simple  geometric  inter- 
pretation. 

Consider  the  family  of  curves 

H^l         (x~ayy  =  M, 

in  the  vicinity  RV*(a). 

The  curve  H^  is  a  hyperbola. 

If  ft<l,  H^  lies  below  H^^  while 
if  /A  >  1,  H^  lies  above  H^  Further- 
more, if  l>ft>)u,',  H^  lies  above  H^.. 
The  curves  H^  all  cut  each  other  at 
the  point  a;  =  a  +  1.  As  we  are  only 
interested  in  these  curves  in  the 
immediate  vicinity  of  the  point  x  =  a^ 
the  point  a-\-l  lies  beyond  the  range 
of  the  figure. 

The  jx  tests  may  now  be  stated  as  follows : 

If  in  some  F'*(a),  |/(aj)|  remains  below  some  H^  which  lies 
below  Hy,  fix)  is  integrable.  If,  on  the  other  hand,  f(x)  has  one 
sign  in  F'*(a),  and  |  f(x)\  remains  above  H^,  the  corresponding 
integral  is  infinite. 


CRITERIA   FOR   CONVERGENCE  409 

583.  Ex.  1.  ^^  ri     dx 

Jo  , 


a/1-x2 

The  only  singular  point  is  x  =  1.     Let  us  apply  the  n  test  at  this  point.     Since 
/(x)  =  -      11  1 


Vl  -  x2      Vl  -  a;      vT+x 
we  see  that  j  ^ 

ilim(l  -x)2/(a;)  =  — . 

We  may  therefore  take  ij.  —  \,  and  J  is  convergent. 

584.   Ex.  2. 


^r 


(?X 


Vx2  _  1 . 1  -  K2a;2 


0</c<l. 


The  singular  points  are  1,    -• 
Consider  the  point  x  =  1. 


\/x2  -  1.1-  /c2x2  =  a/x^^  Vx  +  1  •  1  -  k2x2. 

Hence  i  i 

i21im(x- 1)VW  = 


=1  V2(l  -  »c2) 

In  the  /i  test  we  can  therefore  take  ix  =  h. 

Consider  the  point  x  =  -  • 

As  / 1        \ ^  i 

ilimfi-x)    rCx)  =  —^ , 

x=l  Vk        /  V2(l  -  /c2) 

we  can  take  /u  =  |  at  this  point. 
Thus  J  is  convergent. 

585.    Ex.  3.  r        r^  •         .7  A  ^      ^ 

t/ =  I    logsinxdx,         0<x<ir. 
Jo 
The  singular  point  is  x  =  0. 

We  saw 


Hence 
where 

Thus 
and 
But 

Hence 


lim2i^=l. 

x=0       X 

sin  X  =  xg{x), 
lim  gf(x)  =  1. 

x=0 

log  sin  X  =  log  X  +  log  g(x), 
x^  log  sin  x  =  x^  log  X  +  x'^  log  g{x).,        m  >  0. 
lim  x'^  log  X  =  0,         lim  x^  log  gr(x)  =  0. 

Z=0  2=0 

lim  x^  log  sin  x  =  0. 


Thus  in  the  /x  test,  we  can  take  for  /x  any  positive  number  <  1.      Hence  J  is 
convergent. 


410  IMPROPER   INTEGRALS.     INTEGRAND  INFINITE 

586.   Ex.4. 


J=  f^^^dx,        a>0. 


,  The  singular  point  of  the  integrand  /(x)  is  x  =  0.     In  its  vicinity  F*(0),  /(x) 
has  one  sign  a-  =  +  1. 

Then,  by  579,  J  is  convergent  for  yu,  <  1  ;  and,  by  580,  it  is  divergent  for  m  ^  !• 


587.    Ex.  5 

d  -  \   ■ 


J^  psinx^j.         ^^o_ 

Jo      y-u. 


The  only  singular  point  is  x  =  0.     In  F*(0),  the  integrand  has  one  sign  <r  =  +  1. 

liin5!lL^=l, 

z=l      X 

we  see,  by  579,  that  J  is  convergent  if  /x  <  2 ;  and,  by  580,  that  it  is  divergent,  if  /a  ^  2. 

588.  Logarithmic  tests.  Let  f(^x)  he  regular  in  %  =  (^a.,h^^  except 
at  a.  For  some  M>0,  X  >  1,  s,  let  there  exist  a  V*(a),  such  that 
in  it 

ix-a-)-  ?i-J-  .  ?2-l-  -  ?,_i-i-  •  ts-^  '  \Kx-)\<M.     (1 
x  —  a       x  —  a  x—a       x—a 

Then  f  is  absolutely  integrable  in  21. 

By  389,  5),  for  x>a  sufficiently  near  a,  and  s  =  1,  2,  ••• 

\-l 


DX 


\-K 


1,1  7A         1 


x-a      r^_ay  _j_i I- _ 

x—a     x—a         x—a 

Integrating,  we  have,  for  a  <  a'  <  «"  <  a  +  S, 

p' dx ^    1  r^i-A   1    _  ^i-A   1  1 

Ja     ,  .  ;      1  7A     1  X  — 1|         a"  —  a       *     a'  —  aj 

(x  —  a)l^ •■•  I, —  ^  -• 

X  —  a  X  —  a 

<  e,         for  3  sufficiently  small. 
Thus  the  singular  integral  of  |/(a^)|  at  a  is  evanescent;  for 
J^'\f(ix^\dx<Me. 


CRITERIA  FOR  CONVERGENCE         *  411 

589.    Letf{x)  he  regular  in  %  =  (a,  b')  except  at  a. 
Letf(x)  have  one  sign  a-  in  F^*(a),  where 

(■x-a^l  -1 k^—  •af{x')>M>0. 

X  —  a  X  —  a 

Then  /»& 

J=   1  fdx  =  cr  •  00. 

From  389,  4),  for  s  =  1,  2,  •••,  and  x>a  sufficiently  near  a, 

-»''"■  ^^Ta = z — m — n- ' 


(x—ay^— —  '■■  Is 


X  —  a  X  —  a 

Integrating, 

^ i— =  4+1 '         a<a<c<a-\-6. 

*/a,           xyl  1      ^  a.  —  a 

{x—  a)i^ •••  6, 


X  —  a         X  —  a 
Hence 


\s> 


a—  a 
=  +  00,         when  a  =i  a. 


Hence  ^ 

t/  =s  O-  •  OO. 


590.    The  logarithmic  tests  588,  589  admit  a  simple  geometric 
interpretation. 

Consider  the  family  of  curves 

(J 

Os,k;  y  = i \ — 1— '     >'>i; 


(x  —  a)L 

^          ^  ^x  —  a 

D 

X  —  a 

(x  —  a)L 

.7      1    ' 

and 


in  RV*(a'). 

It  is  shown  readily  that  any  C  curve  finally  lies  constantly  below 
any  D  curve. 


412 


IMPROPER   INTEGRALS.     INTEGRAND   INFINITE 


For  a  given  \,  the  (7  curves  rise  as  s  increases;  while  the  D 
curves  sink  as  in  the  figure. 

The  logarithmic  tests  may  now  be 
stated  as  follows : 

If  |/(a;)  I  finally  remains  below  some 
C  curve,  f{x}  is  integrable.  On  the 
other  hand,  if  /{x}  preserves  one  sign 
near  a,  and  |/(a:)  |  remains  above  some 
D  curve,  the  corresponding  integral  is 
infinite. 


Properties  of  Improper  Integrals 

591.  In  the  following,  as  heretofore  in  this  chapter,  we  shall 
suppose  that  the  integrands  have  but  a  finite  number  of  singular 
points  in  the  intervals  considered. 

When  /(a;)  has  more  than  one  singular  point  in  51  =  (a,  5),  we 
can  break  21  into  partial  intervals,  such  that  /(a;)  has  a  singular 
point  only  at  one  end  of  each  such  interval. 

For  example,  if  the  points  a,  Cj,  c^  are  the  singular  points  of 
/(a:)  in  21,  we  have,  by  576,  577,  /  being  integrable, 

where  aj,  a^  are  points  lying  between      ^ ' 1 1 ^ 

the  singular  points. 

On  account  of  this  property,  we  may  simplify  the  form  of  our 
demonstration  often,  by  supposing  21  to  have  but  one  singular 
point,  which  for  convenience  we  shall  take  at  the  lower  end  of  the 
interval. 


592.    Let  /i(2;),  ■■•/„(2;)  be  integrable  in  (a,  5).      Then 

(^i/i  +  •  •  •  +  c„f„')dx  =  Cj  J^  f^dx  H h  c^J^  f„dx.  (1 

Suppose  /,  •••/„  limited  except  at  a. 


PROPERTIES   OF   IMPROPER  INTEGRALS  413 

Then,  if  a<a<ch, 

Passing  to  the  limit,  we  have  1). 

593.  Let  f(x)  he  integrahle  in  (a,  5).      Then 

f(x')dx=   I  fdx+   I  /<ia;,  a<c<b. 

If  c  is  a  singular  point  of  /,  the  above  relation  is  a  matter  of 
definition  by  577. 

If  c  is  not  a  singular  point,  the  demonstration  follows  at  once 
from  576,  577. 

594.  Jw  21  =  (a,  5)  let  f(x)  he  integrahle,  and 

f(x-)^M. 

Then  r'> 

^J(x)dx^M(h-a'),         a<b.  (1 

For,  suppose  a  is  the  only  singular  point  in  21. 
Then,  if  a<a<b, 

jjdx  ^  M(h  -  a),     by  526,  1. 

Passing  to  the  limit  a=  a,  we  have  1). 

595-    Let  f(x'),  g(x')  he  integrahle  in  (a,  5). 

Except  possihly  at  the  singular  points,  let  f(_x)'>g(x).     Then 

J-'b  r*b 

Jdx>jjdx.  (1 

Suppose  /,  g  are  limited  except  at  a. 
If  a<.a<ih, 

Cfdx>fgdx,     by  526,  2. 

Passing  to  the  limit  a=a,  we  have  1). 


414  IMPROPER  INTEGRALS.     INTEGRAND   INFINITE 

596.  Let  f(x),  g(x)  he  integrahle  in  (a,  h}. 

Except  possibly  at  the  singular  points,  let  f(x')^g(x'). 

At  a  point  of  continuity  c  of  these  functions^  letf{c}  >g(c)'     Then 

j  fdx>  I  gdx.  (1 

Suppose  /,  g  are  limited  except  at  a. 
Let  a<Ca<c<b.     Then,  by  527,  2, 

fdx>  I  gdx; 

a  %y  a 

by  595, 

Adding,  we  have  1). 

597.  Let  f(x)  be  absolutely  integrable  in  (a,  6).     Then  f(pc)  is 
integrable  in  (a,  5},  and 

\£fdx\<£\fcx-)\dx.  (1 

In  578  we  saw  that /(a;)  is  integrable  in  (a,  5). 
Suppose /(a;)  is  limited,  except  at  a. 
Let  a<a<b.     Then,  by  528, 

\Cfdx\<C\f\dx.  (2 

Passing  to  the  limit,  we  have  1). 

Suppose  Cj,  ^21  ■■•  Cs  ^i'6  the  singular  points.     Then,  if  <?«<««< (?,+i, 
fc=  1,  2,  •••  s—  1, 

J'»fr  /»ci  /»ai  /'ft 

Hence 

a  I  It/a      !  It'Ci      I 

.      <  C\f\dx+  r\f\dx+  ...,  by  2), 


<. 


6 

\f\dx. 


PROPERTIES   OF   IMPROPER   INTEGRALS  415 

598.    Let  fix)  he  integrahle  in  21  =  («,  h). 

Let  , 

J=lfdx. 

We  may  change  the  values  of  f(x)  over  any  discrete  point  aggre- 
gate in  %  without  altering  the  value  of  J,  provided  the  new  values 
of  f  are  limited. 

Suppose/ limited  except  at  a.     Let  a<a<b.     Then,  by  530,  2, 


J'^b  nb 

fdx=  I  gdx. 


where  g  is  the  new  function. 

Let  now  «  =  a.     The  integral  on  the  left  converges  to  J.     Hence 


nb  nb 

im  I  gdx  =  I  gdx  =  J- 


1 

f 
599.    1.  Let  f(x)^  ^(^)   ^^   absolutely  integrahle   in   51  =  (a,  6), 
having  none  of  their  singular  points  in  common.      Then  h=fg  is 
absolutely  integrahle  in  21- 

To  fix  the  ideas,  let  c  be  a  singular  point  of  /,  but  not  of  g ; 
a<c<b.     Then  g  is  limited  and  integrahle  in  V^^c'). 

Consider  one  of  the  singular  integrals  of  |  h^x)  |  at  c ;  say  the 
right-hand  one, 

11=    r  \fg\dx,  C<ry'<y"<C-{-S. 

Let  @  be  a  mean  value  of  {gQx')  \  in  F(c).     Then,  by  531, 

R  =  @JJ  \f\dx. 

But  |/(a:)  I  being  integrahle, 

lim  I     \f\dx=0. 

lim  E=0, 
as  @  is  less  than  some  positive  number  M. 


5=0 »-  y 
Hence 


416  IMPROPER  INTEGRALS.     INTEGRAND   INFINITE 

2.  In  %  =  (a,  5),  let  f(x)   he  integrahle  and  g(x)  limited  and 
monotone.      Then  fg  is  integrahle  in  21. 

For  simplicity,  suppose  /  is  limited  except  at  h.     We  must 
show  that 


e>0,         S>0,  \i"fgd 


d 


<  e,  h  —  h<c,  d<h. 


Now,  by  the  Second  Theorem  of  the  Mean,  545, 

^dx  4-  Q(d  —  0^  ^  , 


£fgdx  =  g{c  +  ^)^Jdx  +  gi^d  -  0}ffdx,         c<^<h. 


But  /  being  integrable  in  31,  the  integrals  on  the  right  are 
numerically  as  small  as  we  choose  if  S  is  chosen  sufficiently  small. 

600.  If  f(x)^  Sf(.^^  have  a  singular  point  in  common,  fg  may  not 
be  integrable,  as  the  following  example  shows  : 

f(x)  =  g{x)=        ^       ,        2t  =  (0,  1). 
Vl  -  x^ 

Here,  by  583,  /  and  g  are  absolutely  integrable  in  21.     On  the  other  hand, 

h=fg  =  -^- 
1  —  x^ 

Hence,  by  580,  2, 

\   hdx  =  \   =  +  00, 

Jo  Jo  1  -  x2 

and  h  is  not  integrable  in  (0,  1). 

601.  Letf(pc)  he  ahsolutely  integrahle  in  %  —  (a,  5).  Let  g(x)  he 
iyitegrahle  in  21  and  \g(x)\<Gr.      Then  fg  is  integrable  in  21,  and 

\ffgdx<aC\f\dx.  (1 

For,  g{x')  being  limited  and  integrable,  |  g(^x^  \  is  also  integrable, 
by  507.  Hence  \fg\  is  integrable  in  21,  by  599.  For  simplicity, 
suppose  that  h  is  the  only  singular  point  oi  f(x).     Let  a<^<b. 

Then,  by  528,  \  r^        \       r^ 

1  fgdx\<  I    \fg\dx 

\%J  a  I        %J  a 

<a£\f\dx,  by  529. 

•    Passing  to  the  limit  yS=  6,  we  get  1). 


PROPERTIES   OF   IMPROPER   INTEGRALS  417 

602.    1.  Let  f{x)  he  non-negative  and  integrahle  in  21  =  (a,  6). 
Let  g{x)  he  integrahle^  and 

m<g{x)<M. 
Then  • 

m\   fdx<  I  fgdx  <  M  I  fdx ;  (1 

or  ^ 

\  fgdx  =  g(  fdx,       Cr  =  Meang(x) .     (2 

For,  as  in  601,^  is  integrable.     If,  to  fix  the  ideas,  we  suppose 
h  is  the  only  singular  point  of/,  we  have,  by  529, 


m  j  fdx  <  \  fgdx  <M\  fdx,         a<,fi<.b. 


which  gives  1)  on  passing  to  the  limit  /3  =  h. 

Equation  2)  is  obviously  only  another  form  of  1). 

2.  As  a  corollary  we  have  : 

Let  f(pc)  he  non-negative  and  integrahle  in  (a,  5) ;  while  g(xy  is 
continuous. 

Then  ^  ^ 

I  fgdx  =  g{^)  j  fdx,         a  <  I  <  h. 

3.  By  repeating  the  reasoning  of  534  and  using  596,  we  have : 

Let  f(x~)  he  integrahle  and  non-negative  ;  while  g(x)  is  continuous 
in  %.     Let  c  he  a  point  of  continuitg  off\x^,  and 

Min  g (x')  <g(c')<  Max  g^x),         in  21. 
Then  , 

Jjgdx  =  gCOjjdx,  a<^<b. 

603.    1.  Letf(x)  he  integrahle  in  2t  =  (a,  J).      Then 

fdx,         a,  X  in  %, 

is  a  continuous  function  of  x  in%. 

To  fix  the  ideas,  let  a<a<.x<h. 
Then  ^^ 

AJ=£  fdx. 


418  IMPROPER  INTEGRALS.    INTEGRAND  INFINITE 

If /(a;)  is  limited  in  V(x)-, 

lim  ^J=  0  (1 

ft=0 

by  536.     If  a;  is  a  singular  point,  1)  still  holds  by  574,  since  /  is 
integrable  in  21. 

2.  As  corollary  we  have  : 

Letf(x)  he  integrable  in  (a,  6).      Then 


f(x)dx=  I  f(x)dx,         a<x<h. 

_        a  %/a 


604.    1.  Let  f(x)  he  integrahle  in  %  =  (ah^.     If  f(x)  is  continu- 


ous at  X, 


—  I  fdx  =/(a;),  a,  X  m  21. 

dx^<^ 


To  fix  the  ideas,  let  a  <  a  <  a;  <  5.  Since  /  is  continuous  at  a;,  this 
is  not  a  singular  point  of  /.  Let  c  be  chosen  so  that  a  <  c?  <  a;, 
while  (c,  x)  contains  no  singular  point.     Then  setting 


J'^x  fc  f*x 

a  %/a.  %/c 


we  have  J=  C+K.     But,  by  537, 

dK 


a.   •^<^>- 


Hence,  since  (7  is  a  constant, 

^=  |-((7+  K)=^=f(x). 
dx     dx  dx 

2.  Letf(x)  he  integrahle  in  21  =  (a,  5).     Iff  is  continuous  at  x^ 

-^    r*x+h 
lim-  I      f(x)dx=f(x),         a;  in  21. 

This  is  a  corollary  of  1. 

605.    /w  21  =  (a,  5)  let  /(a;)  5e  integrahle.     Let  it  he  continuous 
except  at  certain  points  c^  ••■  c^,  where  f(x)  may  he  unlimited. 


PROPERTIES  OF   IMPROPER   INTEGRALS  419 

If  F(x)  is  a  one-valued  continuous  function  in  21,  having  f(^x)  as 
derivative,  except  at  the  points  c, 

£fix)dx  =  F(b)  -  ^(a).  (1 

Suppose /(a;)  is  continuous  except  at  a. 
Let  a<a<h.     Then,  by  538, 

£fdx=F(h')-Fia).  (2 

Since  F  is  continuous, 

lim  F{a)  =  F(a'). 


a=a 


Passing  to  the  limit  in  2),  we  get  1). 

Suppose  now  a  and  e  are  points  at  which  /  is  discontinuous.    To 

fix  the  ideas,  let  ^     ^     ^  a 

a<a<c<b. 

T.hen  /»&       /»a       /^c       /»& 

»/a  K/a  \Ja.  %Jc     . 

Now  as  just  shown, 

£  =  Fia)  -  Fia-), 

£  =  F(c~)-F(a-), 

£=F(h-)-F(ic-). 

Adding,  we  have  1)  from  3). 

Ex.  1.  ri      dx  1 

\    — -^— —  =  [arc  sm  x]     =  v. 

Here  i 

/(*)  = 


Vl   -X2 

is  integrable  in  51  =(—  1,  1).     Its  points  of  discontinuity  in  21  are  x  =  ±  1. 

F{x)  =  arc  sin  x 
is  one-valued  and  continuous  in  31,  and  has  /(x)  as  derivative. 
Ex.  2. 


f^=r_ir  =-2. 

J    1X2         L       Xj-i 


This  result  is  obviously  false,  since  the  integrand  is  positive.     The  integrand  is 
not  an  integrable  function  in  (—  1,  1),  by  580. 


420  IMPROPER  INTEGRALS.    INTEGRAND  INFINITE 

Change  of  Vaidable 

606.  1.  Letf{x)  he  in  geiieral  regular  in  %  =  (a^  5)  a^h.     Let 

u  =  <^(a;) 

have  a  continuous  derivative  </>'  (a;)  ^  0,  in  21.  Let  ^  =  («,  /3)  6e  /Ag 
image  of  31,  «w^  Zg^ 

^'6  ^Ag  inverse  function  of  (f).     If  either 

J:c=j^K^)dx,  or  -^„  =  J^/[l/r(M)]l/r'(M)(?W 

z's  convergent^  the  other  is,  and  J^  =  J„. 

By  hypothesis  the  points  of  51,  ^  stand  in  1  to  1  correspondence. 
To  fix  the  ideas,  let  /  be  regular  except  at  a.  Let  <?,  7  be  corre- 
sponding points  in  51,  ^.     Then,  by  543, 

Cfix^dx  =    r^[^(^)]^'(w)c?2^,  (1 

since /(a;)  is  limited  and  integrable  in  (c,  6).  If  now  c  =  a,  7  =  a, 
and  conversely.  Thus  if  either  integral  J^  or  J^  is  convergent, 
the  relation  1)  shows  that  the  other  is,  and  both  are  equal. 

2.  In  ^  =  («,  /S),  «^/3,  let  x^-^Qli)  have  a  continuous  derivative 
which  may  vanish  over  a  discrete  aggregate,  hut  otherivise  has  one 
sign.  Let  51  =  (a,  h)  he  the  image  of  SQ.  Let  f(oc)  he  in  general 
regular  hi  51-     If  either 

is  convergent,  the  other  is,  and  J^  =  J^. 

We  employ  the  reasoning  of  1,  with  the  aid  of  544. 

607.  Ex.  1.  ^^^p    ax 


Vl  -  x''^ 


-*^®*'  X  =  sin  It  =  t/'(i<). 

Here  i/-'  is  positive  in  SB  =(0,  7r/2)  except  at  7r/2. 


SECOND   THEOREM  OF   THE   MEAN  421 

Also  /.^/2  ^ 

Jtt  =  I      du  =  -. 

Jo  2 

Obviously  Ju  is  convergent.     Hence,  by  606,  2, 

Jx  =  t/2, 


a  result  already  obtained. 


Ex.  2. 


dx 


*C2<1. 


V(l   -X2)(l  -k2x2) 

Let 

X  =  sin  M. 

^^     Vl  -  /c2  sin2  u 

But  the  integrand  of  Ju  is  continuous  in  (0,  7r/2)  =  iB.     Hence  Ji,  is  finite. 
Therefore  J^  is. 

Both  these  examples  illustrate  how,  by  a  change  of  variable,  an 
improper  integral  may  be  transformed  into  a  proper  integral. 


Second  Tlieorenn  of  the  Mean 

608.    Let  f(x)  he  integrahle  in  21  =  (a,  5).     Let  gQc)  he  limited 
and  monotone  in  21.      Then 

J=jjgdx  =  gQa  +  ^)£fdx  +  g(h  -  0)f^[fdx,  (1 

where  ^  «.  ^  i 

We  assume  that  g{a  +  0),  g(h  —  0)  are  different,  as  otherwise  1) 
is  obviously  true. 

Let  us  suppose  first,  that  /  is  regular  in  21  except  at  a.     Then,  by 

545,11), 

a<u<b 

Cfgdx^gCa+O^Cfdx  +  K'^}  ■  lg(h-0')-g<ia+0}U       (2 
where  i?(a)  is  a  mean  value  of 

i  fdx,         a<x<h.  (3 


422  IMPROPER   INTEGRALS.     INTEGRAND  INFINITE 

In  2)  let  a  =  a.     We  have 

fdx  =   I  fdx. 

As  all  the  terms  in  2),  except  d(a),  have  a  finite  limit,  it  follows 
that  i?(«)  must  have  a  finite  limit  .y.     Hence 

Cfgdx  =  gia  +  0)  Cfdx  +  ^  5  ^(5  -  0)  -  ^(a  +  0)  I .  (4 

Reasoning  now  as  we  did  at  the  close  of  545,  we  arrive  at 
equation  1). 

Suppose  next,  that  /  is  regular  in  31,  except  at  a,  b.     Then  if 

fygdx  =  g(a  +  0:>fydx+d,(^)\g(:^-0-)-g(a+0}U 

as  we  have  just  seen  in  3). 

Passing  to  the  limit,  we  have,  as  before, 

jjgdx  =  g(a  +  ^~)jjdx  +  ^,\g(h  -  0)-^(a  +  0)|. 

This  may  be  transformed  as  before,  giving  1)  also  for  this  case. 
Let  us  suppose  finally,  that  the  singular  points  of  /  are 

^V   ^2'    '"   '^s' 

Then  ^        ^  ^j 

If  a  or  h  are  singular  points,  the  first  or  last  integral  may  be 
discarded. 

By  the  preceding, 

J"'  =  g(^a  4-  0)j^Jdx  +  ^ (ci -  0)pfdx 

-j^}+K^:-0)|j^  -jj. 


SECOND   THEOREM  OF   THE  MEAN  42^ 

Similarly, 

x;=^(^.+<'){i;'-X}+^«^^-«){X'-X} 

Adding  all  these  equations,  we  get,  setting  e^  =  a,  c^+j  =  b : 
J=  gia  +  0)jr''  +  ^  [g(c^  +  0)  -  gic^  -  0)  |  £^ 

/c=l 

+i;W.+i-o)-^(e,+o)lJ' 

^g(a  +  0)r  +  S+T.  (5 

Now  m,  9}J  denoting  the  extremes  of  the  integral  3), 

mi\g<ic^  +  0)-gic^-0)l<S<W^\gic^  +  0}-g(c^-0)l 
mk]g(c^^,-0-)-gie,  +  0:,l<T<m^lg<:e.-.i-0:f-g(ic^  +  0)\. 

Which  added  give 

m(K^-0)-Ka  +  0))<^+y<aW(^(6-0)-^(a  +  0)). 
Thus  5)  gives 

J=g(a  +  0)  r+  ^Cgib  -  0)-^(a  +  0)\ 

This  is  an  equation  of  the  same  form  as  4).     Thus  reasoning  as 
we  did  on  4),  we  get  1). 


424  IMPROPER  INTEGRALS.    INTEGRAND   INFINITE 

INTEGRALS   DEPENDING  ON  A  PARAMETER 

Uniform  Convergence 

609.  Let  /(a;,  ^)  be  detined  over  a  rectangle  R  =  (a,  5,  a,  /S),  /3 
finite  or  infinite,  and  be  unlimited  in  R.     Let  %=  (a,  6),  ^  =  («,  ^J. 

For  each  y  in  ^,  let  ^^ 

be  convergent.     Then  J"  is  a  one-valued  function  of  y  in  SQ. 

As  in  Chapter  XIII,  we  wish  now  to  study  J  with  respect  to 
continuity,  differentiation,  and  integration,  restricting  ourselves 
to  certain  simple  but  important  cases. 

610.  1.  For  brevity  we  introduce  the  following  terms.  We 
shall  say  f{xy^  is  regular  \w  R  =  {aha^),  /3  finite  or  infinite,  when 

1°.  f(xy^  has  no  points  of  infinite  discontinuity  in  R. 
2°.  f(xy^  is  integrable  in  %  =  (a,  5)  for  each  y  in  ^  =(«,  ;S). 
When  ^  is  finite,  we  shall  sometimes  need  to  integrate  f{xy') 
with  respect  to  y.     In  this  case  we  shall  also  suppose 
3°.  fixy')  is  integrable  in  :^  =  («,  /3)'  for  each  x  in  21. 

For  example,  f(x,  y)  =  y  sin  x  is  regular  in  B. 

If  /3  is  finite,  /  is  limited  in  E.  If  /3  =  oo  ,  /  is  not  limited  in  B  although  it  has 
no  points  of  infinite  discontinuity. 

2.  If  f(xy')  is  regular  in  i2,  except  that  it  may  have  points  of 
infinite  discontinuity  on  certain  lines  a^  =  a^,  •••  a;=a,.,  we  shall 
say  f(xy')  is  regular  in  R  except  on  the  liries  a;  =  a^,  ••• ;  or  that  it  is 
in  general  regular  with  respect  to  x. 

3.  hetf(xy}  be  continuous  in  R  except  on  certain  lines 

x=  «!,  •••  a;=a^;   ?/  =  «p  •••  y  =  «,. 

On  the  lines  a;  =  a^,  •••it  may  have  points  of  infinite  discontinuity  ; 
on  the  lines  y  =  a^---  it  may  have  finite  discontinuities.  If  f(xy') 
is  otherwise  regular,  we  shall  say  it  is  simply  regular  except  on  the 
lines  x  =  a^,  ••■  x=  a^;  y  =  a^^  ..•  y  =  a^;  or  that  it  is  simply  irregu- 
lar with  respect  to  x. 

Thus  .the  smpZy  irregular  functions  are  a  special  case  of  the 
functions  which  are  in  general  regular. 


UNIFORM   CONVERGENCE  425 

4.  The  lines  x  =  a^^  ■■•  x=  a^^  on  which  f(xy^  may  have  points 
of  infinite  discontinuity  are  called  singular  lines. 

The  integrals 

I     fdx^        I       fdx       h  >  0,  arbitrarily  small 

are  called  the  left  and  right  hand  singular  integrals  relative  to  the 
lines  a;  =  a^,  4  =  1,  2  •••  r. 

5.  In  609  we  made  the  formal  requirement  that  fixy')  should 
be  defined  at  every  point  of  R.  It  usually  happens  in  practice 
that  /  is  not  defined  at  its  points  of  infinite  discontinuity.  ;  Such 
is  the  case  in  such  integrals  as 

-\/xy       ^  (x^  +  y^y  ^ 

It  is,  however,  easy  to  satisfy  the  above  requirement  in  all  the 
cases  we  shall  consider;  for,  by  598.  the  value  of 


X/(^'  y)'^' 


is  not  affected  by  a  change  of  the  value  of  /  at  points  lying  on 
the  lines  x  =  a^---  subject  to  the  restrictions  of  that  theorem. 

6.  This  fact  may  also  be  used  to  advantage  sometimes  to  sim- 
plify /(a;,  y)  by  changing  its  value  at  points  lying  on  these  lines. 

611.  1.  Let  f(xy')  be  regular  in  R  =  (aba^'),  /3  finite  or  infinite, 
except  on  x  =  a^,  ••• 

If  the  singular  integrals  relative  to  these  lines  be  uniformly 
evanescent  in  ^,  we  say 

J  =  j^f(x,y}dx 

is  uniformly  convergent  in  ^. 

2.  If  J  is  uniformly  convergent  in  the  intervals  :33i,  •••  ^mi  it  is 
obviously  uniformly  convergent  in  their  sum. 

3.  If  J  is  the  sum  of  several  uniformly  convergent  integrals  in 
®,  it  is  itself  uniformly  convergent  in  ^. 


426  IMPROPER   INTEGRALS.     INTEGRAND   INFINITE 

4.  Letf(xy^  he  in  general  regular  with  respect  to  x  in  Z2  =  (a6a/3), 
^finite.     If  J  is  uniformly  convergent  in  ^,  it  is  limited  in  -©. 

For  simplicity  suppose  x=b  \s  the  only  singular  line.     Then 

fdx\  <  0-,         uniformly  in  ^. 

But  in  the  rectangle  (^ab'a/3), 

\f(xg)\<M. 

\J\<M(h-a^^(T. 


Now  J      ^^'      ^* 

^b 

Hence 


612.    Let  f{xy~)  he  regular  in  R  =(aha^^,  yS  finite  or  infinite^ 
except  on  x=h. 

The  singular  integral  ^^ 

\  fixy)dx,         h,<h'<h,  (1 

is  uniformly  evanescent  in  ^  =  («,  /3)  if 

|/(rry)|<</,(:r),  in  21' =  (Sq,  J), 

and  (f)  is  integrable  in  21'. 

For,  f(xy^   being   limited  and    integrable  in   (5',   5"),   where 
h'  <  h"  <  5,  we  have  for  any  y  m  ^ 

I  Cfdx\<  C\f\dx,     by  528 

<  f  (f>dx,        by  526,  2. 

But  <^  being  integrable  in  21',  we  can  take  6g  so  near  h  that  the 
last  integral  is  <e.     But  as  this  is  independent  of  y, 

f(^xy')dx\  <  e,         in  55. 

b' 

Hence  1)  is  uniformly  evanescent. 


UNIFORM   CONVERGENCE  427 

61 3.   Example.    Let  us  consider  the  integral 

J=  \    log  (1  —  2  y  cos  X  +  y'^')dx  (1 

for  values  of  ?/  in  33  =  (a,  ^),  |3  finite.     The  integrand  f{xy)  is  continuous  except  at 
the  points 

a;  =  m7r,    ?/=(— 1)™,    m  =  0,  ±  1,  ••• 

by  560,  Ex.  2.     Let  us  first  consider  the  singular  integral 

relative  to  the  line  x  =  0.     Since  y  =  1  is  the  only  point  of  infinite  discontinuity  on 
this  line,  we  may  restrict  ourselves  to  an  interval  iB'  =(1  —  tr,  1  +  a).     We  set 


y  =zl  +  h,         \h\<<r. 

Then  .  o      \   ] 

^=j';iog|2(l  +  /.)(l-cosx  +  ^-^)|dx 

=  log2(l+;^)^"dx  +  J;iog(l-cosx  +  ^-^^)^x 

=  «S'l  +   'S^2- 

Obviously  Si  is  uniformly  evanescent  in  58'. 

To  show  that  ;S'2  is  uniformly  evanescent,  we  observe  that 

logf  1  -cosx  H 1   <  |log(l  -  cosx)|, 

since  log  x  increases  with  x,  and  is  negative  for  small  values  of  the  argument.     We 
apply  now  612.     To  this  end  we  show  that 

(t>(x)  =  log  (1  —  cos  x) 

is  absolutely  integrable  in  (0,  a'),  using  the  /^-test. 
Now  in  454,  Ex.  1,  we  saw  that 

J? lim x'^ log (1  -  cos x)  =  0,        0<fi<l. 

Hence  |  0  |  is  integrable,  and  S2  is  uniformly  evanescent.  Hence  >S^  is  uniformly 
evanescent  not  only  in  58',  but  in  33. 

The  same  reasoning  may  be  applied  to  the  singular  integral  relative  to  the  line 
X  =  w.     Here  the  only  point  of  infinite  discontinuity  is  y  =  —  1. 

Hence,  by  611,  2,  the  integral  J  is  uniformly  convergent  in  S3. 


428  IMPROPER   INTEGRALS.     INTEGRAND   INFINITE 

614.  Example.     Let  us  consider  the  integral 

J-\    xv-^  I  log  a;|  Mx,        n  %  0.  (1 

We  show  first  that  it  is  convergent  only  for  y  >  0. 
For,  let  y>0.     Applying  the  /x-test,  we  have 

lim  xi^  -  x^~i  I  log  X I  "  =  lim  x^  |  log  x  | ",        X  >  0 

x=0 

=  0,     by  454,  Ex.  2, 
for  properly  chosen  0  <  m  <  1- 
Hence,  by  579,  J  is  convergent. 

Let  2/^0.     Then        lim a; .  x^-i | log x | »  =  lim x^ ] log x | ",        X^O 

=  +00. 
Hence,  by  580,  J  is  divergent. 

Let  0<  a  <  ^.  We  show  that  J  is  uniformly  convergent  in  35  =(«,  jS).  In  the 
first  place  we  note  that  the  integrand  is  continuous  in  B  —(0,  1,  a,  /3),  except  on 
the  line  x  =  0,  where  it  has  points  of  infinite  discontinuity.  We  have,  therefore, 
only  to  show  that  the  singular  integral  S  relative  to  this  line  is  uniformly  evanes- 
cent.    To  this  end  we  use  612.     Now 

x3'-i|logx|"<  xa-i|logx|",        2/>a. 

But  we  have  just  seen  that 

0(x)=:  X«-l|l0gx|".  (2 

is  integrable.     Hence  S  is  uniformly  evanescent  in  55. 

615.  Let  f(xy)  he  regular  in  M  =  (aJa/3),  /3  finite  or  infinite, 
except  on  x=h.      The  singular  integral 


i 


Jixy^dx,         b'<b, 


is  u7iiformly  evanescent  in  ^  =  (a,  /3),  if 

f(xy^  =  (f)(x)g(xy),     in  R^  =  (h-  8,  h,  ct,  /3) ; 

where  1°  <^(a;)  is  integrable  in  %'  =  (b  —  B,  b')  ; 

2°  g{xy)  is  limited  in  Mf,,  and  integrable  in  any  (b  —  S,  6"), 
b"  <  b,for  each  y  in  Sd- 

For,  by  2°,  l^(a;^)|  <il!f. 

Then  by  1°,  there  exist  for  each  e  >  0,  a  S  >  0  such  that 


IX 


(^(x)dx 


<w  <■' 


CONTINUITY  429 

Then  for  any  y  in  ^, 

fdx\=    I    4)gdx 

Xb" 
<f)dx 

<e,  by  1). 


Continuity 

616.    ] ,  Let  fC^xy')  be  regular  in  R  =  (^aba^},  13  finite  or  infinite, 
except  on  the  lines  x  =  a^,  •  -  •  x  =  a^. 

1°.  Let  the  singular  integrals  relative  to  these  lines  be  uniformly 
evanescent  in  Sd=  («,  yS). 

2°.  Let  rj^  finite  or  infinite.,  lie  in  -53,  and 

lim/(a;y)  =  ^(x)         uniformly 

in  21  =  (a,  5),  except  possibly  at  x  =  a^,  •••  x=  a,.. 
3°.  Let  (fi{x)  be  integrable  in  21.      Then 

Jr'b  ^b   _  r'b 

f(xy')dx=  I    \\mf(xy^dx=  I  (^(x)dx.  (1 

For  simplicity,  we  shall  suppose  there  is  only  one  singular  line, 
viz.  x=b;  we  shall  also  take  ?;  =  oo . 

Let  , 

I>=)  \f(xy)-K^')\dx. 

We  wish  to  show  that 

€  >  0,      (r,     I  i>  I  <  €,        for  any  y>G-. 

Now  .  ,  , 

D=J^  \f(xy')  -  ^(x)  \  dx  +  j^Jdx  -  J^,  <^dx,         b'  <  b, 


430  IMPROPER  INTEGRALS.    INTEGRAND  INFINITE 

Now  by  1°,  3°,  the  last  two  integrals  are  nnmerically  <  e/3,  if  h' 
is  sufficiently  near  6,  for  any  y.  On  the  other  hand,  if  Gr  is  suffi- 
ciently large,  we  have  for  any  y>G-, 

\f{xy^-(\>(x')\< 


for  every  x  in  (a,  J'),  by  virtue  of  2°. 
Hence  |i)j|<e/3.     Hence 

\I)\<e,         y>a, 
which  establishes  1). 

2.  Letf(xy^  he  regular  in  R=  (^aba^'),  ^  Jitiite  or  infinite,  except 
on  the  lines  x  =  a^,  •■■  x=  a^. 

1°.  Let  the  singular  integrals  relative  to  these  lines  be  uniformly 
evanescent  in  SQ  =  («,  /8). 

2°.  Let  7],  finite  or  infinite,  lie  in  ^,  and 

lini/(2:,  ?/)  =  ^(po)         uniformly 

y=r) 

in  %  =  (ab),  except  possibly  at  x  =  a^,  ■■■  x=  a^. 

Then  ^^ 

j  =  lim  I  f(xy^dx         exists. 

3°.  Let  ^(x)  be  integrable  in  21-      Then 

f(xy~)dx=  I    \\m.f(^xy)dx=   |  <^(x)dx. 

We  need  only  show  that  j  exists,  since  the  rest  follows  by  1. 
Let  us  suppose,  to  fix  the  ideas,  that  x  =  b  i^  the  only  singular 
line,  and  that  77  =  00. 

Then 

I)=r\fix,y'^-fix,y"y,dx 

=  £\A^^  y)-/(^,  y")\dx  +  £fix,  y'^dx-£f(x,  y"^dx 
=  L>,  +  D^  +  I)^. 


CONTINUITY  431 

But  by  1°,  there  exists  a  h'  such  that 

|i>2l,     |i>3|<e/4 
for  any  y  in  ^. 

By  2°,  we  can  take  7  such  that  for  any  x  in  (a,  5') 

4(0  —  a) 
Hence 


2(6  -  a) 


for  any  ^',  ?/"  >  7,  and  x  in  (a,  6'). 
Thus  ,  ^  , 

|i>J<6/2. 

Hence 

|X)|<e,         forany?/>7; 

and  the  limit  y  exists. 

617.  1.  In  561  we  have  defined  the  term/(.'r,  3/)  as  a  uniformly 
continuous  function  of  y  in  ^.  It  may  happen  that /(a;,  y  +  h) 
converges  to  f(xy^  for  each  y  in  Sd  and  any  x  in  21,  but  that  the 
uniform  convergence  breaks  down  at  points  lying  on  the  lines 
X  =  a-^^  ••■  x  =  a^.  In  this  case,  we  shall  say/"  is  a  regularly  con- 
tinuous function  of  y  in  ^.  If /(a;,  y  +  K)  converges  uniforml}^  to 
f(xy)  except  on  x=ay,  •••,  where  it  may  not  even  converge  to 
/(a;,  ?/),  we  shall  say  that  f(xy')  is  a  semi-uniformly  continuous 
function  of  y. 

In  both  cases,  we  can  inclose  the  lines  x  =  a^,  •••  in  little  bands 
of  width  small  at  pleasure  but  fixed,  such  that  the  convergence  is 
uniform  in  ^,  when  x  ranges  over  31,  excluding  values  which  fall 
in  the  above  bands. 

2.  It  may  happen  that /(a;,  ?/)  is  a  regularly  or  a  semi-uniformly 
continuous  function  oi  y  in  ^  except  on  the  lines  y  =  a^^  ...  y  =  a^. 
We  shall  say  in  this  case  that  /  is  in  general  regularly  or  semi- 
uniformly  continuous  in  y. 

3.  We  wish  to  make  a  remark  here  which  will  sometimes  per- 
mit us  to  simplify  the  form  of  a  demonstration  without  loss  of 
generality.  In  questions  of  uniform  convergence  or  uniform  con- 
tinuity, the  uniformity  may  break  down  at  points  lying  on  certain 
lines  x  =  ay,  •••  a;=a^.     In  this  case  we  may  count  such  lines  as 


432  IMPROPER  INTEGRALS.     INTEGRAND  INFINITE 

singular  lines.  When  y8  is  finite,  and  no  points  of  infinite  discon- 
tinuity lie  on  these  lines,  their  singular  integrals  are  obviously 
uniformly  evanescent  in  ^. 

618.    1.  As  corollaries  of  616  we  have,  using  611,  4: 

Let  f{xy')  he  in  general  regular  with  respect  to  x  in  R  =(a6a/3), 
/S  finite. 

Let  f(xy^  he  a  semi-uniformly  continuous  function  of  y,  except  at 

Let 


JW  =  i  K^y')dx 


he  uniformly  convergent  in  ;53  =  («/S). 

Then  J  is  limited  in  Sb  and  continuous,  except  possibly  at  a^,  •  •  •  «,„. 

2.   Let  fi^xy)  be  continuous  in  R,  except  on  x  =  a^,   •••  x  =  a^. 
Let  1)  be  uniformly  convergent  in  ^.      Then  J  is  coyitiiiuous  in  ^. 

619.    Ex.  1.  The  integral 

J  =  \    log  (1  —  2  ?/  cos  X  +  y'^)dx 
Jo 

is  a  continuous  function  of  y  in  any  interval  i8  =(a^).  For  the  integrand  is  contin- 
uous in  (0,  TT,  «,  /3),  except  on  the  lines  a;  =  0,  x  =  tt.  In  613  we  saw  J  is  uniformly 
convergent  in  58.     Hence,  by  618,  2,  J  is  continuous. 


Ex.  2.    The  integral 


J  =\  a:^'-!  I  log  2/ |"dx 


is  continuous  in  (a,  /3).        0  <  «  <  ^. 
This  follows  from  618,  2,  and  614. 


Integration 

620.    1.   Up  to  the  present  we  have  been  considering  the  case 
when  the  singular  integrals 

Sc  =jjf(xy)dx 
relative  to  a  line  x=  e  are  uniformly  evanescent. 


INTEGRATION  438 

For  the  purpose  of  integrating 

^(^)  =  )  Kxy)dx 

%J  a 

with  respect  to  the  parameter  y  over  a  finite  interval  ^  =(«,  /3), 
we  can  take  a  slightly  more  general  case. 

As  before,  let /(a;?/)  be  in  general  regular  in  R  =  (^aba^').  To 
fix  the  ideas,  let  us  consider  the  left-hand  singular  integral  at  c. 

Suppose  now  that  for  each  e  >  0  there  exists  a  S  >  0  such  that 
for  any  y  in  Sd*  =  («,  /S*), 

for  every  e'  in  (c—  8,  <?).  Here  o-  is  regular  in  S&  except  at  /9,  and 
is  integrable  in  ^ ;  it  is  also  independent  of  e.  In  this  case  we 
shall  say  this  singular  integral  is  normal.  Similar  remarks  hold 
for  the  right-hand  singular  integral  at  <?. 

2.  Obviously,  if  the  singular  integrals  at  c  are  normal,  they  are 
uniformly  evanescent  in  any  partial  interval  (<^/3')  of  ^. 

Also,  if  the  singular  integrals  at  c  are  uniformly  evanescent  in 
^,  they  are  a  fortiori  normal. 

3.  When  the  singular  integrals  of 


J. 


-■j/(j^y)dx 


are  normal,  we  shall  say  «/is  normally  convergent  in  SQ. 

4.   If  the  singular  hitegrals  at  c  are  normal.,  there  exists  for  each 
e  >  0,  a  S  >  0,  such  that 


\j  "^dyjfda 


<e, 


for  any  c'  in  (c  —  S,  c)  or  (c,  c  -f-  S),  and  any  \,  fi  in  ^. 

To  fix  the  ideas  consider  only  the  left-hand  singular  integral. 
Then 

\)  fd:^ 


ix\<'''^y~^ 


s 


where  _ 

*S'  =  I    ady  >  I    ady. 


Hence 


434 


IMPROPER   INTEGRALS.     INTEGRAND   INFINITE 


I-' 

r/3" 
-/3' 

'1 
a 

821.    Let   f(xy^    he   in   general    regular   with   respect   to   x^    in 
R  =  (aha^^^  /3  finite. 

1°.   Let  f  he  in  general  a  semi-uniformly  continuous  function  of 
y  in  ^  =  («,  /3). 

2°    Let  r* 

he  normally  convergent  in  ^. 

Then  J  is  integrahle  in  ®. 

To  fix  the  ideas  let  x  =  h  be  the  only  singular  line.  Since  the 
singular  integral  is  normal  in  ^,  it  is  uniformly  evanescent  in  any 
(«7),  7  <  /3,  by  620,  2.  Hence,  by 
618,  1,  t/is  integrahle  in  («,  7). 

To  show  that  J  is  integrable  in 
^,  we  have  only  to  show  that 

T=)     Jdy, 

/3-v<^'<^"<^,     

6'   6 
converges  to  9  as  77  =  0. 

T=\     dyj  fdx=j     J    +J     J    =T,+  T,.  (1 

But,  by  620,  4,  there  exists  a  h'  such  that 

|7'2|<e/2,         5'<5, 
however  /3',  /3"  are  chosen. 

On  the  other  hand,  fixy')  being  by  hypothesis  limited  in  any 

(a,  5',  a, /3),         6'<6. 

I  r/^2;|<il!f. 

|«^a  I 

We  can  therefore  choose  77  so  small  that 

|7\|<e/2. 
Hence  1)  shows  that  for  each  e>0,  there  exists  an  17  >0,  such 
that  I  Tl  <  e 

for  any  pair  of  values  yS'^"  in  (/3  —  77,  yS*). 


INVERSION  435 

Inversion 

622.    1.  Let  f(xy^   he  in  general  regular  with  respect  to  x^  in 
R  =  (aba^). 

1°.   Let  the  singular  integrals  he  normal. 

2°.  Let  fixy^  he  in  general  a  semi-miiformly  continuous  function 
of  y  in  Sd. 

3°.   Let  inversion  of  the  order  of  integration  he  permissible  for 
ayiy  rectangle  in  R  not  emhracing  the  singular  lines. 

Then 

K=  \    dy  \  fdx,         ^—  )    ^^ }   f'^y 

exist.,  and  are  equal. 

For  simplicity,  let  us  suppose  a;  =  6  is  the  only  singular  line. 
By  1°,  2°,  and  621,  the  integral  K  exists. 

^J   '^  /»«  ff  fb'  fp 

j    dyj^fdx  =  j^  dxj  fdy.        h-S<b'<b.    (1 
^  |l=j|—  Jl,         since  -£"  exists, 

Hence  1),  2)  give 

I   -  ^  =   )    1  (3 

a     \^ a.  \*J a.   */6' 

<  e,         by  620,  4 

if  S  is  taken  sufficiently  small. 
Now,  by  603,  2, 

j  j     =limj    J    . 

Hence,  by  3),  ^  ^ 

L=      j   =K. 

2.   Letf(xy)  he  simply  irregular  in  R  =  (aha^^  with  respect  to  x 
Let  the  singular  integrals  he  normal.      Then 

£dy£fdx.       £dx£fdy 
exist  and  are  equal. 


436  IMPROPER   INTEGRALS.     INTEGRAND   INFINITE 

This  is  a  special  case  of  1.  For,  by  617,  3,  /  is  in  general  a 
regularly  continuous  function  of  y.  Thus  condition  2°  is  satisfied. 
That  condition  3°  is  fulfilled  follows  from  570,  1. 

623.  Example.     We  saw,  575,  Ex.  3,  that 

^\y~Mx  =  -,        y>0. 
Hence  f  or  0  <  a  <  j3. 

We  can,  by  622,  2,  invert  the  order  of  integration,  since 

\  "xy-hlx,       0  <  a 

is  uniformly  evanescent  in  (a,  ^)  by  614. 
Thus  1)  gives,  inverting, 

log  ^  =  (\lx  (^xy-'^dy  =  C  ^^'^  ~  ^""^  dx-  '  (2 

a      Jo      Ja  Jo        log  X 

For  a  =  1,  this  gives 

f'^^~'-^dx  =  log^.  (3 

JO        log  X  ^ 

If  we  set  here  /3  =  2,  it  gives 

r^^dx  =  log2.  (4 

Jo  logx 

624.  We  give  now  an  example  where  it  is  not  permitted  to 
invert  the  order  of  integration. 

We  have  for  all  points  different  from  the  origin, 

x^  +  y'^     (x^  +  y'^y 

D        y      ^    x-^-y2 
*x2  +  y'^      (x2  +  2/2-)a 
Thus 

Jo      ^Jo(x2  +  2/2)2  Jo      ■'[x^+lJ^So        Jo   1+2/2        4' 

Jo      Jo(x^  +  y-2yi    "  Jo       \x-^  +  yij>         Jol+a;2         4 

Hence  A,  B  are  both  convergent ;  but  they  are  not  equal. 


INVERSION  437 

625.  1.  Up  to  the  present  we  have  supposed  that  the  points  of 
infinite  discontinuity  of  the  integrand  f^xy)  He  on  certain  lines 
parallel  to  the  «/-axis. 

We  consider  now  a  more  general  case. 

Let  us  suppose  that  these  points  of  infinite  discontinuity  do  not 
lie  only  on  a  finite  number  of  lines  parallel  to  the  z/-axis,  but  that 
it  is  necessary  to  employ  in  addition  a  finite  number  of  lines  par- 
allel to  the  a;-axis. 

To  fix  the  ideas,  let  these  lines  be 

x  =  a^,  •••  x  =  a,.',    y=a^,  .-'  y=a,.  (1 

If  f(xy')  is  otherwise  regular,  i.e.  if  properties  2°,  3''  of  610,  1 
hold,  we  shall  say  f(xy^  is  regular  in  R  except  on  the  lines  1),  or 
that  it  is  in  general  regular  with  respect  to  x,  y. 

2.  Similarly  we  extend  the  term  simply  regular,  viz. :  If  /(xy') 
is  continuous  in  R  except  on  a  finite  number  of  lines  parallel  to 
each  axis,  say  the  lines  1),  where  it  may  have  points  of  finite  or 
infinite  discontinuity ;  if,  moreover,  it  enjoys  properties  2°,  3°  of 
610,  1,  we  shall  say  f(xy^  is  simply  regular  in  R  except  on  the 
lines  1),  or  that  it  is  simply  irregular  with  respect  to  x.,  y. 

3.  The  lines  y  =  a^.,  •••  are  also  called  singular  lines,  and  the 
integrals 


r>y 


are  singular  integrals  relative  to  the  lines  y  =  a^,  tw  =  1,  2  •••  s. 

4.  In  accordance  with  the  present  assumptions,  we  should 
modify  the  definition  of  normal  singular  integrals  given  in  620,  1, 
so  as  to  allow  o-(«/)  to  have  singular  points  at  a^,  •••  a^. 

As  an  example,  consider 


V(x-&)(^-/3) 


Here  every  point  on  the  lines  x  =  b,  y  =  ^  are  points  of  infinite  discontinuity. 

If  E  =(«,  b,  a,  ^),  a<Cb,  a<C^,  we  see  at  once  that /is  continuous  in  B  except 
on  the  above  lines.  Obviously,  /  is  integrable  with  respect  to  x  or  y  in  B.  Thus  / 
is  simply  regular  in  B  except  on  the  lines  x  =  b,  y  =  ^. 


438  IMPROPER   INTEGRALS.     INTEGRAND   INFINITE 

626.  1.  Lt^t  f(xy)  he  regular  in  R  =  (aha^')^  except  on  the  lines 
:r  =  (/p  •••  x  =  a,.;  y  ^  n^.  ■■■  y  =  (^s- 

1°.  Let  it  he  in  general  a  semi- uniformly  continuous  function  of  y 
in  ^  =  (  «/3). 

2°.  Let  the  singular  integrals  relative  to  the  lines  x  =  a^^  •■•  he 
nor/nal  in  SQ 

^°.  Let  the  singular  integrals  relative  to  the  lines  y  =  a^^  ■■•  he 
uniformly  evanescent  iii  any  interval  of  %  —  (^ab),  not  emhracing  the 
points  ttj,  •••  a,,. 

4°.  Let  any  integral         £dy£fdx 

admit  inversion  if  the  rectangle  (a'h'  a' ^'^  does  not  embrace  one  of  the 
singular  lines. 

^^^^^  Jiy):=Cfix,y-)dx 

%/  a 

is  integrahle  in  ^. 

For  simplicity,  let  us  assume  that  there  is  only  one  singular  line 
x=h,  and  one  line  y  ='  /3. 

We  observe,  first,  that  J  is  integrable  in  any  («/3')i  /3'  </3  by 
1°,  2°,  and  620,  2,  621.  Therefore,  to  show  that  J  is  integrable 
in  Sd,  it  suffices  to  prove  that,  for  each  e  >  0  there  exists  an  t;  >  0, 
such  that  ^^" 

is  numerically  <  e  for  any  pair  of  numbers  /3'/3"  in  (5  =  (^  —  97,  /3). 
T=XdySjdx=X£^X,X=T,+  T,.  (1 

But  by  2°  and  620,  4,         ,  ^  ,  ^    /^ 

if  h'  is  taken  sufficiently  near  h.     Suppose  b'  so  chosen  and  then 
fixed. 

On  account  of  4°,  ^  ^  T*'  r^\ 

By  virtue  of  3°  we  can  take  7;  >  0  sufficiently  small,  so  that 

^^         I      2(6  —  a) 


INVERSION  439 

"^"^"  I^il<e/2. 

Therefore  \T\<€,         for  any  /3'/3"  in  (5. 

2.   ie^  f(xy^  he  simply  regular  in  R  =  (aba^^,  except  on 
x=a^,  ■■■  «/=«!,  ••• 

1°.   Let  the  singular  integrals  relative  to  x=  a^,  •■■  he  normal  in  ^. 
2°.   Let  the  singular  integrals  relative  to  g  =  a^,  •••  be  uniformly 
evanescent  in  any  interval  of  §1  not  embracing  the  points  a^,  •  •  • 
Then 

is  integrahle  in  ^. 

For,  conditions  1°,  2°,  3°  of  1  are  obviously  fulfilled. 
That  4°  is  satisfied,  follows  from  570. 

627.    1.  Let  f(xy^  he  regular  in  R  =  (aha^^  except  on  the  lines 

x=  a^,  ■■■  x  =  a,.;  y=a^,  ■••  y=a^. 

1°.   Let  the  singular  integrals  relative  to  the  lines  x  =  a^^  •••  he 
normal. 

2°.  Let  the  integral  T'^t    C^fj 

admit  inversion,  if  (c,  d)  does  not  embrace  the  points  a^,  ag,  ••• 
3°.  Let 

K=\  dy\  fdx 

he  convergent. 

hThen  L-Mydy 

■is  convergent,  and  K=  L. 

For   simplicity,   let   x=b,  y  =  ^   be    the   only  singular   lines. 
Then,  by  2°, 

jjxj^dy  =j%£fd^^       h-8<h'<h.  (1 

=  1     I  —I     }   ■,       since  ^  is  convergent. 


440  IMPROPER   INTEGRALS.     INTEGRAND   INFINITE 

But,  however  small  e>0  is  taken,  we  may  take  S>0  so  small  that 

irX'h^'     by  620,  4.  (3 

From  1),  2),  3)  we  have 

I  C^'  C^         I 

-  Jrke,  b-S<b'<b. 

|«/a    %/ a 

But  then  ^^,  ^^  ^6-  r^ 

L=\     \  =  lim  =  JT. 

2.  Let  fixy)  he  simply  regular  in  R  =  Qaba/3^,  except  on  x  =  a^  ••-, 
y  =  «i  ••• 

1°.  Let  the  singular  integrals  relative  to  the  lines  x  =  a^  ■•■  be 
normal  in  ^. 

2°.  Let  the  singular  integrals  relative  to  the  lines  y  ■=  a-^  •■•  be  uni- 
formly evanescent  in  any  interval  of  2t  not  embracing  the  points  a^  ■■• 

Then  the  integrals 


£dy£fdx,      £dx£fdy 


are  convergent  and  equal. 

For  condition  2°  of  1  is  satisfied  by  622,  2  if  we  replace  a:  by  ^ 
in  that  theorem.     Condition  3°  is  fulfilled  by  626,  2. 

628.    Example.     As  an  application  of  627,  2  let  us  consider 

r%  r^,    6, /3>o. 

The  singular  lines  are  aj  —  0,  2/  =  0. 
The  singular  integral  relative  to  a;  =  0, 


Jo  , 


dx 
is  normal  in  53  =  (0,  /3).     For, 

Vy*^"  Va; 
setting 

J"  Vx  y/y 


DIFFERENTIATION  441 

Here,  for  any  e  >  0,  there  exists  a  5  >  0  such  that  e'  <  e,  for  any  0  <  a  <  S.     On 

the  other  hand,  a-  is  integrable  in  53. 

The  singular  integral 

j,^  r°  dy 

relative  to  y  =  0  is  uniforiuly  evanescent  in  any  (a,  6),        a  >  0.     For, 

~Vrt^"  y/y 
But  for  any  e  >  0  there  exists  an  Uo  >  0,  such  that 

pi^^eVa,        0<«<ao. 

Hence  T<.  e        for  any  « <  ao. 

Thus  the  conditions  of  627,  2  being  fulfilled,  both  integrals 

Jo       Jo  ^—  Jo       Jo  ^ 

are  convergent  and  equal. 

This  result  is  easily  verified  by  actually  evaluating  these  integrals.     For, 


"  Vxy     Vy*^*'  Vx       Vy 


Hence 


Pr  =  2y^P^  =  4V6^;etc, 

Jo  Jo  Jo   Vm 


Vy 


Differentiation 

629.   Xg^  f{^l/^i  f'y(j^y^  ^^  regular  in  R  =  (^aha^')^  except  on  the 
lines  x=  a^  ■■■  x  =  a^. 

V.   Letf'y  he  a  semi-uniformly  contiviuous  function  of  y  in  ^  =  («/3). 
2°.   In  the  elementary  rectangles  (a^  —  r,  a^  +  t,  a,  /8), 

|/;(a:y)|<<^X^),         i  =  l,  2,  ...  r, 
the  (/)/s  6em^  integrahle  in  (a^  —  S,  a^  +  S). 


442  IMPROPER   INTEGRALS.     INTEGRAND   INFINITE 

For  simplicity,  let  x=b  be  the  only  singular  line,  and  let fy  be 
uniformly  continuous  in  21  =  (a,  6),  except  at  b.     Then 

A?/     ^«  h  '    ' 

=  j  f'yix,  rj'ydx,         by  Law  of  Mean 

=  Cfyixy^dx  +  D.  (2 

\fy(xr^~)-fyixy-)\<^4>iix'), 
\D,\<2Jjdx<'-, 


But 

Also,  by  2= 
Hence 


provided  h'  is  taken  sutBciently  near  h. 

On  the  other  hand,/^  being  uniformly  continuous  in  21'  =  (a,  6'), 
we  can  take  h  so  small  that 

\f'yix,  v)  -fy{^.  y~)  I  <  2(5^'       i^  21'. 
Then  lAK^A 

Hence  |Z)|  <  e,  for  any  |  A|  <  8  ;  and  1)  follows  at  once  from  2). 

630.    Example.     As  an  application  of  629,  let  us  consider  the  integral 

I  a;*'  -1  log"  xdx^        n  ^  0,  integral, 

wliich  was  taken  up  in  614.     The  integrand 

f(xy)  =  a^-i  log"  X 
is  not  defined  for  x  =  0.     Let  us  give  it  the  value  0,  when  x  =  0.    Then 

/^(x,  ?/)  =:xs'-Uog"+ix        forrc>0 
=  0  for  X  =  0. 


DIFFERENTIATION  448 

Then  /  and/y  are  simply  regular  in  R  =  (0,  1,  a,  /3),  «>0  except  on  the  line 
X  =  0.  Moreover,  fy  is  a  uniformly  continuous  function  of  ?/  in  58  =  («,  /3),  except 
possibly  on  the  line  x  =  0.  It  is  therefore  a  regularly  continuous  function  of  y  ni  ^. 
Thus  condition  1°  of  629  is  fulfilled.  Condition  2°  is  also  satisfied,  as  614,  2)  shows. 
Hence  if  we  set 


J 


=j;x.-idx, 


we  get,  by  629,  ,  ^       -i 


dy 


or  differentiating  n  times, 


dy 


T       /^i 

-=  I  xJ'-Uog«xdx.  (1 

»      Jo 


But,  by  575,  Ex.  3,  , 

J  =  -- 


Hence 


y 

^=(_1)«_!LL.  (2 

ayn  yn+1 


Comparing  1),  2),  we  get 

rx3'-ilog''xrfx=(-l)"-^,        y>0.  (3 

Jo  2/»+i 

631.  1.  By  using  double  integrals,  we  can  obtain  more  general 
conditions  than  those  given  in  629,  for  differentiating  under  the 
integral  sign  in 

\J  a 

For  example,  the  following. 

Let  f(x^  y')^  f'y(x^  y^  he  in  general  regular  with  respect  to  x  in 
Ji  =  (aba0). 

1°.  Let  f'y{xy^  he  a  semi-uniformly  continuous  functioyi  of  y  in 
«  =  («, /3). 

2°.  Let 

he  uniformly  convergent  in  S8. 
3°.  Let 

Xy+fi         /»6 
dyj/I^Cx,  y)dx,         \h\<8 

admit  inversion.  ^ 

Then  , 

f^=j/lCxy}dx,         in«.  (1 


444  IMPROPER  INTEGRALS.    INTEGRAND   INFINITE 

For 

=  i J^  dxj^     f'yix,  y)dy,         by  605,  610,  6 
1    ry+h 

=  U     Ky)'iy-  (2 

But  by  1°,  2°,  and  618,  1,  g(^y}  is  continuous  in  SB.     Thus  on 
passing  to  the  limit  h  =  0,  in  2),  we  get  1),  using  537,  2. 

2.  As  a  corollary  of  1  we  have : 

Let  f(xy')^  fy(xy')  be  regular  in  Il  =  (aba^'),  except  on  the  lines 
x=  ay,  ••'  x  =  a^. 

Letfy(xy')  he  continuous  hi  R,  except  on  these  lines. 
Let 

he  uniformly  convergent  in  ^  =  (a,  ^S). 
d    C  C 


CHAPTER   XV 
IMPROPER  INTEGRALS.    INTERVAL  OF  INTEGRATION  INFINITE 

Definitions 

632.  1.  If  f(x)  has  no  points  of  infinite  discontinuity  in 
21=  (a,  00 ),  and  is  integrable  in  any  partial  interval  (a,  5),  of  21, 
A'e  shall  say  that  f(x)  is  regular  in  21.  If  on  the  contrary,  /(a:) 
has  a  finite  number  of  points  of  infinite  discontinuity  Cj,  Cg,  •••  in 
21,  but  is  integrable  in  any  partial  interval  (a,  J),  we  shall  say 
f(x}  is  in  general  regular  in  21-  The  points  Cj,  c^^  •••  are  singular 
points. 

Let /(a;)  be  in  general  regular  in  21-     Let  us  consider 

lim  j  f{x)dx^  (1 

a=cc»' a 

which  we  denote  more  shortly  by 


I  f{x)dx, 

^  a 


(2 


and  which  is  called  the  integral  of  f(x)  from  a  to  +  oo,  or  the 
integral  of  f{x)  in  21. 

If  the  limit  1)  is  finite,  we  say  the  integral  2)  is  convergent  or 
that /(re)  is  integrable  in  21.  If  the  limit  1)  is  infinite,  the  inte- 
gral 2)  is  divergent.  If  the  limit  1)  does  not  exist,  i.e.  if  it  is 
neither  finite  nor  definitely  infinite,  the  integral  2)  does  not  exist. 

If  |/(a;)  I  is  integrable  in  2t,  /(a^)  is  absolutely  integrable  in  21, 
and  the  integral  2)  is  absolutely/  convergent. 

2.  We  make  a  remark  here  which  will  often  enable  us  to  sim- 
plify our  demonstrations,  without  loss  of  generality. 

445 


446  INFINITE   INTERVAL   OF   INTEGRATION 

If /(a;)  is  in  general  regular  in  51=  (a,  oo),  we  can  take  h  so 
large  that /(a;)  is  regular  in  (5,  oo).     But  the  integral 

j  f{x)dx 

*y  a 

has  been  treated  in  Chapter  XIV.     We  have  thus  only  to  consider 

I  f{x)dx, 

in  which  the  integrand  is  regular.  We  may  therefore  often  as- 
sume in  our  demonstration,  without  loss  of  generality,  that  fix) 
is  regular  in  51- 


Ex.  1. 
For, 

Ex.  2. 

Eor, 

Ex.  3. 
For, 

does  not  exist. 


C^  dx 

\     —  =  1  ;        it  IS  convergent. 

iC=ooJl    X?  \  X] 

J—  =  +  CO  ;        it  is  divergent. 
1      X 

lim  \    —  =  lira  log  x  =  +  oo . 

x=xJ'^     X 

I    cos  X  dx        does  not  exist. 
Jo 

lim  \    cos  X  dx  =  lim  sin  x 

X — 00  */ 0 


633.    1.    G-eneral  a-iterion  for  convergence. 
Letf(x)  he  regular  in  %—  (a,  cx)).     For 


to  he  convergent,  it  is  necessary/  a7id  sufficient  that,  for  each  e  >  0, 
there  exists  a  G->0,  such  that 


£fdx\<e  (2 


for  any  pair  of  numhers,  a,  ^^  Gr. 
This  is  a  direct  consequence  of  284. 


DEFINITIONS  447 

2.  The  integral  2)  is  perfectly  analogous  to  the  singular  inte- 
grals considered  in  Chapter  XIV.  It  is  convenient  to  call  it  the 
singular  integral  relative  to  the  point  a;  =  oo ;  and  also  to  call  this 
point  a  singular  point. 

Instead  of  2)  it  is  convenient  at  times  to  call 

ffix-ydx,      b>a,  (3 

the  singular  integral.     The  integrals    2)  and    3)    are   obviously 
equivalent. 

3.  Letf(x)  he  regular  in  %  =  (a,  oo).  Iff(x)  is  absolutely  inte- 
grahle  in  21,  it  is  also  integrahle  in  %. 

For  by  hypothesis 

\    \f(ix')\dx<e,  G<a<l3. 

But  in  any  given  (a,  ^),f(x)  is  integrable  by  hypothesis  and 

I  rfdx\<  r\f\dx,         by  528,  1. 

4.  The  reader  should  note  that  an  integral  may  be  convergent 
and  yet  not  absolutely  convergent.     Thus 

r^^^dx 

-^0        x'' 
converges  for  0  <  ^u,  <  1,  while 

Jo        of 
is  divergent  for  these  values  of  /x.     Cf.  646,  Ex.  2. 

634.    1.  The  symbol 

^J(x)dx  •  (1 

is  defined  as 

lim   1  f{x)dx^ 

and  is  called  the  integral  of  f{x)  from  —  oo  to  a. 


448  IXFINITE   INTERVAL   OF   INTEGRATION 

The  symbol 

rf(x)dx  (2 

is  defined  as 

lim    I  f(^x)dx. 

The  terms  introduced  in  632  have  a  similar  meaning  here. 
The  integral  1)  is  not  essentially  different  from 


fix)dx, 

and  requires  therefore  no  comment. 

2.   In  regard  to  2),  we  have  the  theorem : 

Let  f(pc)  b&  in  general  regular  in  (—  oo,  oo).     In  order  that 

J=  I   fdx 

be  convergent.,  it  is  necessary/  and  sufficient  that 
^1  =  J   fdx,   J^  =  J^  fdx 
be  convergent  for  any  a.      When  J  is  convergent^ 

If  J  is  convergent,  we  have 

€>o,  ^>o,  \j-  r|<J,  «<-(7,  /3>a. 


Subtracting,  we  get 


■       -te|<6. 


I  *J  a.     I         ^ 

\J  a.     I 
a 


DEFINITIONS  449 

Hence,  by  633, 

J   fdx 

is  convergent.     Similarly, 

I  fdx 

is  convergent.     Therefore  when  J  is  convergent,  Jy,  J^  are  con- 
vergent. 

Conversely,  if  J^,  J2  ^^^  convergent,  J  is  convergent. 

For,  let  a  be  fixed  and  u<a<  ^.     Then 

r=  r+  C\     by  598. 

*/(z  »/a  */a 

For  a  sufficiently  large  (r,  and  e  >  0  arbitrarily  small, 
Hence 


<e. 


J     -(^1  +  ^2) 
for  all  a  <  —  (r^,  and  /3  >  (3^.      Therefore 

lini  j     =  Jj  +  t^2- 
Hence,  when  Jj,  J2  '^'"6  convergent,  J  is,  and 

3.  As  a  result  of  the  foregoing,  we  see  that  the  integrals 

Cfdx.  J  fdx  (3 

do  not  differ  essentially  from 

C/dx. 

For  convenience,  we  shall  therefore  study  only  this  last;  the 
results  we  obtain  are  then  readily  extended  to  the  integrals  3). 


450  INFINITE   INTERVAL   OF   INTEGRATION 

Tests  for  Convergence 

635.  1.    The    /*   tests  for  convergence.     Let  f(x)   be   regular  in 
51  =  (a,  oo).     If  there  exists  a  /m  >  1,  such  that 

x>^\f(x')\<]yL    M>0,         inVCcc); 

f(x)  is  absolutely  integrable  in  St. 

For  let  (7  <  a  <  /3  lie  in  V.     Then,  by  526,  2, 

<  e,  if  Cr  is  taken  sufficiently  large. 

2.  Let  f{x)  be  regular  in  %  =  (a,  oo).     If  for  some  /jl>1 

lim  x'^\f(^x^)\ 

is  finite.,  fi^^  *'s  absolutely  integrable  in  21. 

3.  As  corollary  of  2  we  have : 
In  %  =  (a,  oo),  a  >  0,  let 

f(x)  =  ^-^^^  or      "^      ^ ,        X.>1;   t'>0, 

where  g  is  limited  in  21  and  integrable  in  any  (a,  6).      TAg/i  /(^c)  is 
absolutely  integrable  in  21- 

636.  Test  for  divergence.     Let  f(x')  be  regular  in  2t  =  (a,  oo). 
In  F^(oo)  let  f  have  one  sign  o-,  and 

axf{x)>M,  M>0. 

Then  ^^ 

J=  \    fdx  =  cr  •  oo. 

For,  let  a<a<ix;  while  «,  2;  lie  in  V. 
Tlien 

»/a  «/a  */a 

Now,  by  526,  2, 

Cafdx>M  T— =  iJf  lofT  -  =  +Q0, 
when  a;=  +  cc. 


TESTS   FOR   CONVERGENCE  451 

637.    Logarithmic  test  for  convergence.     Let  f(x)  he  regular  in 
31  =  (a,  oo).     If  there  exists  a  /i>l,  and  an  s,  such  that 

xl-^xl^x  ■  ■  ■  l^_ixl/x Ifix') I  <  iHf,         m  Fi^QO ), 

/(a;)  is  absolutely  integrable  in  21. 

We  have,  by  389,  2),  for  x>0  sufficiently  large, 


DM-'^x  = 


l-f. 


Hence,  if  0  <  (r  <  «  <  /3, 

^a   xl^x---l,>^x      /L4  — lU/-'a      ^/~^/3j 

when  (r  =  +  00. 
Now 

I  fdx\<\     \f\dx<MJ    — ^^ 


Li'x 


<  e,  for  G-  sufficiently  large. 


638.    The  logarithmic  test  for  divergence.     Let  f(x)  be  regular  in 
5t  =  («,  oo).     In  T'X^)'  ^^^  /('^)  have  one  sign  a;  and 

xljxl^x  ■■■  l^x  ■  <TfQc) >  iHf  >  0. 

Then  ^^ 

\    fdx  =  a  ■  cc. 

*'^^'  /^X  /»a  /^X 

•>^u  *^a  ^  a. 

j    afdx>MJ      j""  J     =  Ml,,,~ 


since  by  389,  1),  for  sufficiently  large  a;  >  0, 
As 

X 

lim  ls+i-  =  +  Q0> 


our  theorem  is  establisherl. 


452  INFINITE   INTERVAL   OF   INTEGRATION 

639.     Ex.  1.    A  quarter  period  of  Jacobi's  function  sn(u,  k)  is 


r       ^^  — ■    o<.<i. 


K--  

\/(l-x--i)(l-K2a;2) 

The  point  x  =  l//c  is  tlie  only  finite  singular  point.  Applying  the  jx  test  of  579 
at  this  point,  we  see  that  we  can  take  ij.  =  \.  The  singular  integral  at  this  point  is 
therefore  evanescent.     Consider  now  the  point  x  =  oo.     We  have 

1  1 


V(l-x2)(l-/c2a;2) 


W(-i.)(-^.) 


This  shows  that  the  /u  test  of  635  is  satisfied  for  any  /i<2.     Hence  the  singular 
integral  relative  to  this  point  is  evanescent.     Hence  K  is  convergent. 

640.     Ex.  2.  A  half  period  of  Weierstrass's  function  p(«,  $^2,  gz)  is 

dx 


■  r 


^74  x^  -  g-zx  -  gs 
where  ei  is  the  largest  real  root  of 

4  x'5  -  g-zx  -  gz. 

The  point  ei  is  the  only  finite  singular  point.  Applying  the  /u  test  of  579  at  this 
point,  we  see  that  we  can  take  p.  =  \.  Hence  the  singular  integral  relative  to  this 
point  is  evanescent.     Consider  the  point  x  =  oo.     We  have 

1  1 


X^ 


This  shows  that  the  ix  test  of  635  is  satisfied  for  any  ^^■%\.    Hence  w  is  convergent. 


641.     Ex.3.    The  Beta  function, 


B  (m,  v)  =  r  _^li^_.  (1 

^        ^     Ju    (l  +  x)«+«'  ^ 

The  point  x  =  0  is  a  singular  point  if  m  <  1 .     Applying  the  tests  of  679,  580  at  this 
point,  we  see  that 

I    7       o>0, 

>  (l  +  x)«+" 

is  convergent  when  tt>0;  and  divergent,  when  m^O.     Consider  the  point  x  =  oo 
We  have 

•^^^'^  ~  (1  +  x)»+'' ~  x^+i  *  /        l\«+v' 


TESTS   FOR   CONVERGENCE  453 

Applying  now  the  tests  of  635,  636,  we  see  that 

r  f{x)dx,        M  >  0. 

converges  for  v>0,  and  diverges  for  v^O. 

Thus  the  integral  1)  has  a  finite  value  for  every  «,  u>0.     The  function  so  de- 
fined is  called  the  Eulerian  integral  of  the  first  kind  or  the  Beta  function. 

642.     Ex.4.    The  Gamma  function, 

r(u)  =  ij    e-^x»-^dx.  (1 

The  point  x  =  0  is  a  singular  point  if  m  <  1. 
Applying  the  tests  579,  580  at  this  point,  we  see  that 


f        a>0. 

Jo  , 


is  convergent  when  ii  >  0,  and  divergent  when  m  .^  0. 

On  the  other  hand,  applying  the  ix  test  of  635,  3,  we  see  that 


r 


is  convergent  for  any  u.     The  integral  1)  therefore  defines  a  function  of  u  for  all 
?/  >  0.    It  is  called  the  Eulerian  Integral  of  the  second  kind  or  the  Gamma  function. 

643.    1.   Let  f{x)  he  regular  and  integrahle  in  any  partial  interval 
(a,  A),  of%=  (a,  oo). 

Let  fh^'ydx 

be  limited  zw  21. 

In  ^(oo),  let  g(x)  he  monotone,  and  ^(oo)  =  0. 

Then  f{x)g(x)  is  integrahle  iti  21. 

We  apply  the  criterion  of  633.     Let   (r<«<y3  lie  in   F(oo). 
Then  by  the  Second  Theorem  of  the  Mean,  545, 

f%dx  =  g(a  +  0:>f^fdx+g{^  -  0)f^fdx,         «<  ^<.i3. 

We  can  take  Gr  so  large  that 

are  numerically  as  small  as  we  please.     As  the  integrals  on  the 
right  are  numerically  less  than  some  fixed  number,  we  have 


I  r^ 

I  fgdx 


<e 

for  any  pair  of  numbers  a,  ^>Q-.     Hence  ^  is  integrahle  in  %. 


454  INFmiTE   l^^TEKVAL   OF   INTEGRATION 

2.  Letf^x)  he  regular  and  integrahle  in  %  =  (a,  oo).  Let  g(x)  he 
limited  and  monotone  in  21. 

Thenfg  is  integrahle  in  51. 

For,  by  545, 

Jjgdx  =  g(a  +  0)fjdx  +  g(/3  -  0')ffdx,  (1 

Let 

\g(x-)\<M. 

Since  /  is  integrahle  in  51,  we  can  take  y  so  large  that 

Iffdxl        IffdxU-^;  «, /3>7. 

Then  the  right  side  of  1)  is  numerically  <  e.  Hence  fg  is 
integrahle. 

3.  Let  f(x^  he  regular  and  absolutely  integrahle  in  51=  (a,  oo). 
Let  g(x)  he  limited  and  integrahle  in  51.  Then  f(x)g{x)  is  ahsolutely 
integrahle  in  51. 

We  have  only  to  show  by  633  that 

e>0,  6?>0,         j]fg\dx<e,  (1 

for  any  pair  of  numbers  a,  ^>  Cr. 

Now  g(^x}  being  limited,  we  have  ' 

|^(a:)|<il!f,         in  51. 
Hence 

r\fg\dx<Mr\f\dx.  (2 

%/a  *^a 

But  /(a;)  being  absolutely  integrahle  in  51,  we  can  take  Gr  so 
large  that 


f]f\dx<j^.       u,^>a. 


M 
This  in  2)  gives  1). 

644.    Let  f(x)  he  in  general  regular  in  51  =  (a,  oo),  hut  not  inte- 
grahle in  5t.     Let 

)Kx)dx  (1 


TESTS   FOR   CONVERGENCE  455 

he  limited  in  51.     Let  g(x')  he  monotone  in  %  and  ^(oo)=(r^0. 
Then 

fj{x)gix)dx  (2 

18  not  convergent. 

For,  if  2)  were  convergent,  we  would  have 

I  C^ 
e>0,  h,  I  fgdx  <e,  a,  ^>b. 

\%/  a. 

But  this  is  impossible.     For 

£fgdx  =  gCa  +  0}£fdx  +  ^(/3  -  Q-yf^'/dx.  (3 

Let  the  integral  1)  be  numerically  <  M. 
We  can  take  h  so  great  that 


\g(x)-a\<(7,         x>b. 


where  a-  is  small  at  pleasure. 
We  can  therefore  write  3) 


f%dx  =  (G-\-  <^')f[fdx  +  (  a  +  a"yfydx.  \a'l   |o-"  |  < 


Hence 


Jf'^^'-^l  <  — TTTi —  <  ^''       ^  small  at  pleasure, 

which  states  that /(a;)  is  integrable  in  %. 

645.   Ex.  1,    For  what  values  of  fx  does 

j^  pcosx^^ 

Jo       ^f. 

converge  ? 
Set 

j^^C''92^ax,        j,=  C  92^dx;       a>0. 

Jo        ^M  Ja        x'^ 

The  integral  Jo  is  convergent  by  643,  1,  provided. /u > 0.    For  ix30,  it  obvioualy 
does  not  converge.    The  integral  Ji  is  convergent,  as  we  saw,  586,  only  when  /u<l. 
Thus  J  is  convergent  when  and  only  when 

0</U<l. 


456  INFINITE   INTERVAL   OF   IN  rEGKATION 

646.   Ex.2.  ^„  . 

j=C  !HL^^x. 

Jo       y.^ 

Set 


Ji—  \    dx,        J'2=  \      dx;        fl>0. 

Jo       j-M  Ja.        y^fx. 


In  587,  we  saw  Ji  is  convergent  only  when  m  <  2. 

By  643,  1,  J2  is  convergent  when  ;u  >  0.     When  ,u  ^  0  it  obviously  does  not  con- 
verge. 

Hence  J  is  convergent  when,  and  only  when, 

0<M<2. 
That  J  does  not  converge  absolutely  for  0  <  /x  <  1,  is  shown  as  follows.     We  have 

Jo  ^i^  Jo  Jn  J(n-l)TT 

=  Jl^-J-2+--+Jn. 

x  =  y  +  (m  -  l)Tr. 

p        sinydy        >  ^L_  f  sjn  y  c?y  = -2_. 
Jo  {y  +  (m  -  l)ir}'^       (witt)"  Jo  (mTr)'* 


Let 
Then 


Hence 


T-  ^  2  r  1  ,  1   ,       ,11. 

A„> —  J ^  =00, 


when  n  =  00,  as  the  reader  probably  knows,  or  as  will  be  shown  later. 

Properties  of  Integrals 

647.    In  Chapter  XIV   we    established    the   properties   of   the 
improper  integrals,  ^^ 

fdx. 


by  a  passage  to  the  limit.  We  propose  now  to  develop  the 
properties  of  improper  integrals,  the  interval  of  integration  being 
infinite,  by  a  similar  method.  In  many  cases  the  reasoning  is  so 
similar  to  that  employed  to  prove  the  corresponding  theorems  in 
Chapter  XIV,  that  we  shall  not  repeat  it,  referring  the  reader  to 
the  demonstrations  given  in  that  chapter. 

648.    1.  Let /(x)  he  integrahle  in  (a,  oo).     Then 

fdx  =  -jydx. 


PROPERTIES   OF   INTEGRALS  457 

2.  LetfQc)  he  integrahle  in  (a,  oo).      Then 

I  fdx=  I  fdx->r  I  fdx,         a<h. 

3.  Letf-^(x')  •••fmQ'O  ^^  integrahle  in  (a,  oo).      Then 

^  (p\f\  +  •  •  +  Cmfm)dx  =  c J^it^aj  +  ■  •  •  C;J^ A<^a;. 

649.  1.  Letf(x)^  g{x)  he  integrahle  in  (a,  oo).  Except  possibly 
at  the  singular  points  let  f(x)^g(x).      Then 

Jfdx>  i  gdx. 
a  %/a 

2.  Letf(x)^  ^(^)  he  integrahle  in  (a,  oo).  Except  possihly  at  the 
singular  points,  let  f(x')  ^g{x^.  At  a  point  c  of  continuity  of  these 
functions,  letf(c)>g(^c).      Then 

£fdx>J^  gdx. 

3.  Let  f{x)  ^0  be  integrahle  in  (a,  oo).  At  a  point  c  of  continuity 
offletf(ic)>0.      Then 

j^f{x)dx>0. 

4.  Let  fix)  he  absolutely  integrahle  in  (a,  oo).      Then 

\fjd.\<£\fid.. 

5.  Let  ^oo 

j=lfd. 

he  convergent.  We  may  change  the  value  of  f(x)  over  a  limited  dis- 
crete aggregate,  ivithout  altering  the  value  of  J,  provided  the  new 
values  off  are  limited. 

650.  Let  f(x)  be  integrahle  in  21  =  (a,  oo).     Then 

Jr»ao 
fdx         a<  x. 
X 

is  a  continuous  limited  function  of  x  in  31. 


458  INFINITE   INTERVAL   OF  INTEGRATION 

For,  by  648,  2,  ^^      ^^      ^„ 

1=1+1,         c>x. 

*^x  *^x         *^c 

But 

j  fdx 

is  a  continuous  function  of  x  in  (a,  c),  by  603.     As 

fdx 

is  a  constant,  J(x)  is  continuous  in  21. 

J  is  limited  in  %.     In  fact,  for  each  e  >  0,  there  exists  a  c  such 
that 


I  ff'^- 


'a^  <  e. 


But 

being  continuous  in  the  limited  interval  (a,  c),  is  limited.     Hence 
J  is  limited  in  21. 

651.  Let  f(x)  he  integrahle  in  21  =  (a,  oo).     Then 

for  any  point  x  of  %^  at  which  fix")  is  continuous. 

For,  ii  e>x,  ^  ^ 

J(x)=i     =1    +(     =K(x-)+0, 

^x  *^x  *^c 

C  being  a  constant.     By  604,  1, 

dK  j.^  >. 

ax 

Hence  dJ^d{K±0)^dK^,,.. 

dx  dx  dx 

652.  In  21  =  (a,  oo),  let  f{x^  he  continuous  excepting  possibly  at 
certain  points  c^---  c^,  where  it  may  he  unlimited.  Let  it  he  inte- 
grahle in  any  (a,  ?»). 


THEOREMS   OF   THE  MEAN  459 

Let  F(x)  he  one-valued  and  continuous  in  21 ;    having  f{x)  as 
derivative  except  at  the  points  c. 
Then 

f  fdx  =  F(+oo-)-F(a-),  (1 

where  F(  +  oo)  is  finite  or  infinite. 

For,  by  605,  , 

I  fdx^^Fih-y-FQa-), 

however  large  h  is.     Passing  to  the  hmit,  we  get  1). 

Theorems  of  the  Mean 

653.  First  Theorem  of  the  Mean.     In%  =  (a,  oo)  let  gQic)  he  inte- 
grable  and  limited. 

Let  f(x)  he  integrahle^  and  non-negative  in  %.      Then 

rfgdx=mrfdx,  (1 

where  2)^  is  a  mean  value  of  g  in  21. 
For,  by  602, 

m\  fdx<\  fgdx^M)  fdx,         a<h.  (2 

^vhere  ^    r  \  ^  tvt 

m<g(x)<M. 

Let  5  =  +  00  ;  since  all  the  integrals  in  2)  are  convergent,  by 
643,  3,  we  get  in  the  limit, 

m  r  fdx<  I    fgdx<M\    fdx, 
which  gives  1). 

654.  Second  Theorem  of  the  Mean.     In  %  =  (a,  oo),  let  f(x)  he 
integrahle  and  g(xy  limited  and  monotone.      Then 

J=  Cfix^gix^dx  =  g(a  +  0)  f^fix^dx  +  ^(o))  CfCx^dx,      (1 

a<r)<oo. 


460  INFINITE   INTERVAL   OF   INTEGRATION 

If  ^(«-l-0)=^(+ go)  the  theorem  is  obviously  true.  We  may 
therefore  assume  that  these  limits  are  different.  Next  we  observe 
that  the  integral  J  is  convergent,  by  599,  2  and  643,  2. 

To  fix  the  ideas,  let  g^x}  be  monotone  increasing. 

Let  b  be  arbitrarily  large.     Then,  by  608, 

Cfgdx  =  ff(a  +  0)  Pfdx  +  gib  -  0)  Cfdx.  (2 

Let  us  add  _„ 

j5  =  K+<»)(   fdx 

to  both  sides  of  2);  observing  that 

lim^  =  0.  rS 

We  get 

ffgdx  -\-B  =  g(ia  +  Q)  f^fdx  +  g(b  -  0)  Cfdx  +  ^(  +  oo)  Cfax 

^a  ^a  »^f  *^b 

=  ^(«  +  0)jr7^a;  -  ^(a  +  ^~)f fdx  +  ^(5  -  0) J/^a; 
-  9Q>  -  0)  rV^a;  +  ^(  +  00)  Cfdx 

^h  ^b 

=  ^(«  +  ^)fydx  +  p(6  -  0)  -  ^(a  +  0)  |/7^a; 

+  {K  +  ^)-^(^-0)|jr7(^rc 

=  ^(a  +  0)  J  /c?x  +  ^7+  V.  (4 

Let  X,  /i  be  the  minimum  and  maximum  of 

fdx 
in  21.     Then  obviously, 

fdx  <  /*, 


J ^00 
I    fdx<fjk. 
b 


THEOREMS  OF   THE   MEAN  461 

Hence 

f^(5-0)-K«+0)}\<C^<l^(5-0)-^(a  +  0)|/i, 

Adding,  we  get 
]g(^+co')-g(^a+Q)\\<U+V<\gi  +  co')-gia^O^\,i. 

Hence  ?/+ r=  9)?'1^(+ oo)-^(a  + 0)1,  (5 

where  ^      ™,, 

From  4)  and  5)  we  have 

rfgdx  +  B  =  g{a  +  0)rfdx  +  m'\g(i+^')-gia  +  Qy,. 

Passing  to  the  limit  h  =  co,  and  using  3),  we  have 

J=:  g(a  +  0)fjfdx  +  m  |^(  +  o))  -  ^(a  +  0)  I ;  (6 

6=00 

and  -      cYw 

\<^<fi. 

The  integral  ^oo 

J,  -^^^ 

being  a  continuous  function  of  x  in  (a,  oo),  must  take  on  the  value 
'3R  for  some  point  x  =  r),  finite  or  infinite,  in  this  interval.     Then 

m  =  Cfdx. 

From  6)  we  have  now, 

J=  g{a  +  ^^fjdx  +  |^(  +  «))  -  ^(a  +  0)  ]^ffdx 

=  g(a  +  0)  Vfdx  +  ^(  +  oo)  f  /(^a;, 
which  is  1). 


462  INFINITE    INTERVAL   OF   INTEGRATION 

Change  of  Variable 

655.    Let  ^  =  («,  /S),  a^  /3;  either  a  or  ^  may  he  infinite. 
Let  X  =  i/r(w)  have  a  continuous  derivative  in  ^,  which  may  vanish 
over  a  discrete  aggregate.,  hut  has  otherwise  one  sign. 
Let  %  =  (a,  6)  he  the  image  of  ^,  where 

a  =  lim  yjr(u),   h  =  lim  '^(u)  ; 

and  h  may  he  infinite. 

Letf(x)  he  integrable  in  any  (a,  J'),         h'  <  b. 
If  now,  either 

J^=J^f(x)dx,  or  J^=  J^f[y\r(u)']y\r'(u)du 

is  convergent,  the  other  is,  and  hoth  are  equal. 

There  are  various  cases.  Let  us  take  the  following  as  illustra- 
tion. Let  21  =  («,  oo),  ^  =  (a,  oo).  By  virtue  of  403,  the  points 
of  21  and  ^  are  in  1  to  1  correspondence. 

Let  b',  /8'  be  corresponding  points  in  21,  ^.  Then  as  yS' =  co, 
5'  =00  also,  and  conversely. 

By  606,  2,  ,  , 

j   f(x)dx  =  (   f[ylr(iu}-]ylr'(u-)du.  (1 

Suppose  Jj.  is  convergent.  Then  1)  shows,  passing  to  the  limit, 
that  J^  is  convergent  and  Jj.  =  J^.  The  supposition  that  J^  is  con- 
vergent leads  to  a  similar  conclusion  for  J^.. 


656.   Ex.  1. 

Consider  the 

convergence  of 

J-- 

-  \    sin  x2  dx. 

Jo 

Since  the  intej 

grand 

is  continuous, 

,/  converges  if 

J.- 

=  r  sinx^dx 

converges.     Let 

X  - 

=  \p{u)  —y/u  ; 

and 

2t 

=  (1, 

Od),     33  =(1,00 

) 

Then 

/•=c  c,;.,  ,, 

which  is  convergent  by  646. 

Hence,  by  655,  Jx,  and  therefore  J,  is  convergent. 


CHANGE   OF   VARIABLE  463 

Ex.  2.     We  found,  by  630,  3),  that 


X' 


cc3'-i  log"  a;  d«  =  ^ — •^^""'•,        y>0.  (1 

Let  us  set 

z  =il/(u)  =  e-". 

Here 

0  =  0,     6  =  1;      a=+oo,     /3  =  0. 

In  58, 

is  continuous  and  always  negative. 

Then,  by  655,  /-o 

J'a  =  -  ( -  1)"  I    e-'^yu'^du  (2 

is  convergent,  since  1)  is.     Hence  1),  2)  give 


r 


e-^^M"  dtt  =  -^i^ ,        2/  >  0.  (3 

yn+l 


657.    1.  Stake's  Integrals.    Let  us  consider  the  convergence  of  the  integral 

J'=  I    X  &\n  {x^  —  xy)dx,,  (1 

which  comes  up  in  the  theory  of  the  Rainbow. 

Let  us  set 

u  =  x^  -^  xy  =  x(x2  —  ?/)  =  (^(x).  (2 

The  graph  of  this  is  a  curve  which  crosses  the  axes  at  the  points  x  =  0,  x  =  ±  \/y, 
if  ?/  >  0  ;  and  at  the  point  x  =  0,  if  y  =  0.  To  fix  the  ideas,  let  us  suppose  ?/  >  0  ; 
the  case  when  y  ^0  may  be  treated  in  a  similar  manner. 

Supposing,  therefore,  y>0,  the  graph  of  2)shows  that  as  x  rangesover  2I=(Vy,  ao), 
u  ranges  over  i8  =  (0,  oo),  the  correspondence  between  the  points  of  31  and  58  being 
uniform.     Thus  the  relation  defines  a  one-valued  inverse  function  x  =  f  (m)  in  SB. 

Let  us  write  1) 

j=  (''+  r, 

Jo  Jy/p 

and  denote  the  latter  integral  by  J^. 

The  corresponding  integral  in  m  is  ^ 

J"„  =  I   g(u)  sin  u  du, 

setting 

9(u)  =       "" 


3x2-?/ 


We  can  now  apply  648,  1.  For,  x  =  +  oo  as  w  =  +  oo.  Hence  g(u)  is  a  monotone 
decreasing  function,  for  any  positive  y,  and  gf(ao)  =  0.  Thus  Ju  is  convergent. 
Hence  Jx  is  ;  and  therefore  the  integral  J  is  convergent. 

2.   The  same  considerations  show  a  fortiori,  that 

ir=  r  cos  (x^  —  xy)dx  (3 

is  convergent  for  any  y. 


464  INFINITE   INTERVAL   OF   INTEGRATION 

3.    In  connection  with  these  integrals,  occurs  another  integral 

L=  i  x"^  cos (x^  —  xy)dx,  (4 

which,  it  is  important  to  show,  is  not  convergent.     In  fact,  effecting  the  change  of 
variable  defined  by  2),  in 

Lx=  \  X-  cos  (x3  —  xy)dx, 

supposing  to  fix  the  ideas  that  y  >  0,  we  get 

r°°      x'  C^ 

i„  =  1    cos  udu=  \  Mil)  cos  u  du. 

Jo  3x2-2/  Jo    ^  ■' 

Here  k{u)  is  a  monotone  function,  and 

Thus  Lu  is  divergent,  by  644.     Hence  ix  is.     Therefore  L  is  divergent. 
INTEGRALS  DEPENDING  ON   A  PARAMETER 

Uniform  Convergence 

658.    1.   Let  f(x^  y)  be  defined  at  each  point  of  the  rectangle 
B,  =  (aooa/8),  /3  finite  or  infinite.      Let  %  =  (a,  oo),  :53  =  («,  /5). 
We  shall  say  f{xy~)  is  regular  in  R  when  : 

1°.  f(xy')  has  no  point  of  infinite  discontinuity  in  R. 
2°.  fQcy')  is  integrable  in  51  for  each  y  in  ^. 

At  times  Ave  shall  need  to  integrate  f(xy^  with  respect  to  y. 
In  this  case  we  shall  also  suppose : 

3°.  f(xy')  is  integrable  in  ^  for  each  x  in  31. 

2.  If  f(xy^  is  regular  in  R,  except  that  it  may  have  points  of 
infinite  discontinuity  on  certain  lines  x  =  a^^  ■■■  x=  a^,  we  shall  saj^ 
f{xy^  is  regular  in  R  except  on  the  lines  x=  a^,  •••  or  that  it  is  in 
general  regular  with  respect  to  x. 

3.  Let  us  suppose  that  the  points  of  infinite  discontinuity  of 
f(xy^  do  not  lie  all  on  a  finite  number  of  lines  parallel  to  the 
y-axis,  but  that  it  is  necessary  to  employ  in  addition  a  finite  num- 
ber of  lines  parallel  to  the  a:;-axis.  To  fix  the  ideas,  let  these  lines 
he  x=  ay,  •••  x=  a^\  y  =  a^,  •■■  y  =  ag.  If  f(xy^  is  otherwise  regu- 
lar in  jB,  i.e.  if  it  enjoys  properties  2°,  3°  of  658,  we  shall  say 


UNIFORM   CONVERGENCE  465 

f(xy^  is  in  general  regular  with  respect  to  x,  y,  or  f(xy^  is  regular 
except  on  the  lines  x=  a^^  •••  y  =  a^,  ••• 

4o  Let/(a;«/)  be  continuous  at  each  point  of  R  except  on  certain 
lines  rr  =  aj,  •  •  •  a;  =  a^ ;  y  =  a^^  -••  y  =  a^.  On  the  lines  x=  a-^^  •••  it 
may  have  points  of  infinite  discontinuity ;  on  the  lines  y=a^,  •  •  • 
it  may  have  finite  discontinuities.  If  f{xy^  is  otherwise  regular 
in  a,  we  shall  say  it  is  simply  regular  with  respect  to  x  except  on  the 
lines  x  =  a^,  ■••  or  that  it  is  simply  irregular  tvith  respect  to  x. 

5.  Let  f(xy')  be  continuous  at  each  point  of  R  except  on  the 
lines  2;=  a  J,  •••  x=a,,;  y  =  a.^,  •••  y=a^.  As  in  3,  let  us  suppose 
that  all  the  points  of  infinite  discontinuity  cannot  be  brought  on 
the  lines  x  —  a^^  •■■  \jetf(xy^  be  otherwise  regular.  We  shall  say 
fQxy)  is  simply  irregular  with  respect  to  x,  y^  or  that  it  is  simply 
regular  except  on  the  lines  x=  a-^^  •••  y  =  a^^  •  •  • 

6.  The  lines  x  =  a^^  ••■  y  =  a^,  •■■  on  which  are  grouped  the 
points  of  infinite  discontinuities  oif(xy^^  are  called  singular  lines. 
To  each  of  these  belong  right  and  left  hand  singular  integrals  as 
in  Chapter  XIV.     Cf.  666. 


7.  The  integral 


f{xy')dx,        x^Gr,  (1 


where  Gr  is  large  at  pleasure,  is  called  the  singular  integral  relative 
to  the  line  x  =  cci. 

If  for  each  e>0  there  exists  a  Gr,  such  that  1)  is  numerically 
< e  for  any  y  in  ^  and  every  x'^Gr,  we  say  1)  is  uniformly  evanes- 
cent in  ^. 

8.  If  the  singular  integrals  relative  to  the  lines  x=a^,  ••-  x  =  a,., 
as  well  as  the  singular  integral  relative  to  the  line  iC  =  oo  are  uni- 
formly evanescent  in  ^,  we  say  the  integral 

J  =  £f(x,y~)dx  (2 

is  uniformly  convergent  in  ^. 

If  the  uniform  convergence  of  J  breaks  down  at  certain  points 
Yj,  •••  7i  in  ^,  we  shall  say  J  is  in  general  uniformly  convergent 
in  ^.     Cf.  666. 


466  INFINITE    INTERVAL   OF   INTEGRATION 

9.  As  in  Chapters  XIII,  XIV,  we  wish  now  to  study  the  inte- 
g-ral  2)  with  respect  to  continuity,  differentiation,  and  integra- 
tion. We  may  often  simplify  our  demonstrations  without  loss  of 
generality  by  observing  that  we  may  write 

fdx  =   I  fdx  +  J  fdx  =  t/j  +  J^. 

Here  we  may  take  h  so  large  that  none  of  the  lines  a;  =  aj,  -• 
fall  in  (JooKyS). 

The  integral  J^  has  been  treated  in  Chapter  XIV. 

10.  In  this  article  we  have  considered  f(xy)  chiefij^  with  respect 
to  X.  Evidently  we  may  interchange  x  and  ^,  which  will  give  us 
similar  definitions  with  respect  to  y. 

We  wish  also  to  note  that  all  the  following  theorems  apply  to 
the  integral 

on  interchanging  x  and  y. 

659.  Let  f(xy^  he  regular  in  R  =  (acca^^,  y3  fi7iite  or  infinite. 
Let  ^(x)  he  integrahle  in  51,  and . 

1/(2^3/)  I  <</>(^).         in  R. 
Then  ^^ 

j    /(^,  y)^^  (1 

is  uniformly  convergent  in  ^. 

For  ,      ,„  ,„ 

I     fdx\<  I      \f\dx,         by  528. 

<  f  Mx,         by  526,  2. 
Since  <p  is  integrahle  in  51,  we  can  take  h  so  large  that 

I      (f)dx  <  e 

b' 

for  any  pair  of  numbers  5',  h"  >  h. 

Hence  1)  is  uniformly  convergent  in  ^. 


UNIFORM    CONVERGKNCE 


467 


660.    1.  Letf(xy^  he  regular  in  R=  {acca^),  ^finite  or  infinite. 

Let 

fix,  i/)  =  (t){x')g(x,  y), 

where.,  1°,  ^  is  absolutely  integrahle  in  51.      2°,  g(xy^  is  limited  in  R 
and  integrahle  in  any  (a,  h},for  each  y  in  SQ. 
Then 

ffixy^dx  (1 

is  uniformly  convergent  in  ^. 
For,  g  being  limited  in  i2,  let 

\g{xy)\<M. 
By  1°,  there  exists  for  each  e  >  0,  a  5  such  that 

jj'\4>ix')\dx<^.         b<b'<b",  (2 

Then  for  any  y  in  ^, 

fdx  =    I     (l>gdx 

I    b'  I      \*yb' 

<MC\(l>\dx,        by  529. 

<6,         by  2). 
Hence  1)  is  uniformly  convergent  in  Sd- 

2.  As  corollary  of  1,  we  have,  by  635 : 

In  R=  (^acca^^;  a  >  0,  ^finite  or  infinite,  let 


where  g  is  limited  in  R,  and  integrahle  in  any  (a,  5)  for  each  y  in  ^. 
Then 

1    fi^y^dx 

is  uniformly  convergent  in  ^. 


468  INFINITE   INTERVAL   OF   INTEGRATION 

661.    1.  Let  f(xy)  he  regular  in  R=  (aooayS),  ^  finite  or  infinite. 

where  1°,  <^(x)  is  integrahle  in  21.     2°,  g(xy^  is  limited  in  M  and  a 
monotone  function  of  x  for  any  y  in  ^. 
Then  ^ao 

1    f(xy^dx 

is  uniformly  convergent  in  ^. 

For,  b}^  the  Second  Theorem  of  the  Mean,  545, 

r  (l)gdx  =  g(h' +  0,  2/)  P4>dx^g(h"-Q,y^  C  cf>dx,     h<h' <^<h" . 
*/()■  ^b'  *^f 

But  g  being  limited  in  i2,  and  ^  integrable,  the  right  side  is 
numerically  <e  for  any  ^  in  ^,  and  any  pair  of  numbers  J',  5", 
provided  h  is  taken  large  enough. 

2.  Letf{xy^  he  regular  in  R  =  (aQoa/3),  ^  finite  or  infinite.     Let 

where  1°,  h(xy^  is  limited  in  R  and  monotone  for  each  y  in  Sd, 
and  2°,  ^^ 

)    9{^y')dx 

^  a 

is  uniformly  convergent  in  ^. 
Then  ^^ 

I  fixy')dx 

is  uniformly  convergent  in  ^. 
For,  by  545, 

r  fdx  =  h(h'  +  0,  y')  f^gdx  +  h(h"  -  0,  y)  C  gdx.  (1 

*^b'  *^b'  -^$ 

But  h  being  limited  in  R, 

\h(xy-)\<M. 

On  the  other  hand,  by  2°,  there  exists  a  h^  such  that  each  integral 
on  the  right  of  1)  is  numerically  <  e/2  M  for  any  pair  of  numbers 
h\h">b^. 


UNIFORM  CONVERGENCE 
Hence  for  any  y  in  48, 


469 


<€. 


662.    Integration  by  Parts.     Letf{xy^  he  regular  in  J?=(aQ0a/3), 
^finite  or  infinite.      The  integral 

^=  (    Axy')dx 

is  uniformly  convergent  in  SQ,  if 

J    A^'<y)dx=F{x,y)-^J    g(x,y^dx;  (1 

where  both  expressions  on  the  right  are  uniformly  evanescent  for 

a;=  oo. 

For  then,  for  each  e  >  0  there  exists  a  b  such  that 


^(^,  y') 


< 


<l 


(2 


for  any  a:>5,  and  any  y  in  ^. 
Hence  1)  and  2)  give 

\ffdo. 


<€. 


663.    Examples. 
1.  The  integral 


•« = X' 


g_X^_l    ^y.^ 


(1 


defining  the   Gamma  function,    considered   in  642,    is   uniformly  convergent    in 

g3  =  (a,  ^),        «>0. 

For,  consider  the  singular  integral  relative  to  x  =  0. 

We  have,  since  0  <  x  <  1, 

x3'-i  <  xi-i,        in  S. 

Hence  ,  „  i  ^       i 

Thus,  by  612,  the  singular  integral  relative  to  x  =  0  is  uniformly  evanescent  in  S5. 
Consider  next  the  singular  integral  relative  to  x  =  co.     We  have,  since  x  >  1, 

e-xa3,-i^5f_  in  SB. 

^    e^ 

Hence  this  singular  integral  is  uniformly  evanescent  in  33.    Thus  1)  is  uniformly 

convergent  in  33. 


470  INFINITE   INTERVAL   OF   INTEGRATION 

This  is  uniformly  convergent  in  any  33  =  («,  qo),  which  does  not  contain  the 
point  ?/  =  0,  as  may  be  seen  by  662.     For,  integrating  by  parts, 

Jx         X  L     xy     Jj       Jx        x^y 

_  COS  xy      p  cos  xy  ^^  ^2 

xi/        J^      x'^y 

To  fix  the  ideas,  suppose  a  >  0  ;  then 

I  COS  xy  I  ^  1 
I    xy    I      ax 

This  shows  that  the  first  term  on  the  right  of  2)  is  uniformly  evanescent  in  S3. 
The  second  term  is  uniformly  evanescent  by  660,  1,  as  is  seen,  setting 

.  /  N       1         /     N      cos  x?/ 
x^  y 

For  later  use,  let  us  note  that 

y=x     xy  i/=x  y  Jx       x? 

lim  J'=0. 


y=«     xy  y=«  y 

Hence 


r^ 


-  sin  Xx  dx  (3 

is  uniformly  convergent  in  53  =  (0,  oo),  by  661.     For,  in  the  first  place,  the  inte- 
grand/(xy)  is  continuous  in  i2  =  (0,  oo,  0,  co).     For,  the  only  possible  points  of 
discontinuity  lie  on  the  line  x  =  0. 
But,  the  Law  of  the  Mean  gives, 

e~'y  =\  -xy  +  '^^^  e-O'v. 
2! 

Hence  for  x  ^  0,         ^^^^^  ^  ^  ^.^  ^^  _  ^^2^-0x3,  gin  Xx,        0  <  ^  <  L 

This  shows  that  /  is  continuous  at  each  point  on  the  y  axis,  if  we  give  to  /  the 
value  0  at  these  points. 

This  fact  established,  we  can  apply  661  by  setting 

^(a;)=5HL2^,   g(^xy)=\-e-'y. 

Then  (/>  is  integrable  in  (0,  00  )  by  646  ;  while  g  is  obviously  'limited  in  S,  and  a 
monotone  increasing  function  of  x  for  each  y  in  S.  Hence  3)  is  uniformly  conver- 
gent in  S3. 


UNIFORM  CONVERGENCE  471 

4,  r""  sin  XM  cos  Xa;  ,  .. 

\ dx  (4 

Jo  X 

is  uniformly  convergent  in  33  =  (0,  qd)  except  at  y  =  |  \  |. 

For,  in  tlie  first  place  the  integrand  /(x,  y)  is  continuous  in  B  =  (OfloOco)  if  we 
give  to /the  value  y  at  the  point  (0,  y). 

For,  the  Law  of  the  Mean  gives 


Hence  for  x^O, 
Thus 


sin  xy  =  xy ^sinOxy,        0<tf<l. 


f{xy)  =  y  cos  Xx  — |-  sin  exy  cos  Xx. 
lim/(x,  y  +  h)-y. 

x=0,  h=0 


This  established,  we  have  only  to  show  that  the  singular  integral 

B  =  Cjlxy)rix,         b<,b'<  b", 
is  uniformly  evanescent  in  SB. 

Now  by  the  Second  Theorem  of  the  Mean,  545, 

B  —  —\    smxycos\xdx-\ I     sin  xw  cos  Xx  dx.  (5 

b'Ji-  b"h 

But  for  2/=^  I X  I, 

C  •  >     ,  cos  (y  —  X)      cos  (y  +  X)  ,„ 

Jsmx,cosXxdx=-^^iLi__yL_Z.  (6 

Let 

\y-\\,        \y  +  \\><7.  (7 

Then  6)  shows  that  each  of  the  integrals  in  5)  is  numerically  <2/<r. 
Let  therefore,  &>4/e(r;  then 

l-B|<e, 

for  any  y  satisfying  7).     That  is,  B  is  uniformly  evanescent  except  at  y=|\|. 
Hence  the  integral  4)  is  uniformly  convergent  except  at  this  point. 
We  may  arrive  at  this  result  more  shortly,  making  use  of  2. 
For, 

2  sin  xy  cos  Xx  =  sin  x  (2/  +  X)  +  sin  x  (y  —  X)  • 
Hence 

'  sin  xy  cos  Xx  /**  sin  x  (y  +  X)        ,   f*  sin  x  (y  —  X) 


r°°  sm  xy  cos  Xx       _  /**  sin  x  (y  +  X)  /"* 

Jo  X  Jo  X  Jo 


X 


dx. 


Here  the  first  integral  is  uniformly  convergent,  if  j/  :jb  —  X ;  the  second  integral 
is  uniformly  convergent,  ify^\. 


r°°  X  sin  xi 

Jo       1  +  X2 


dx 


is  uniformly  convergent  in  (a,  co),  a>0. 

For  .  .  - 

X  sm  x?/  _  sm  xy        1 

1  +  X2    ~        X        ',    .    1' 

We  have  now  only  to  apply  661,  2,  using  the  result  obtained  in  Ex.  2. 


472  INFINITE  INTERVAL   OF  INTEGRATION 

^-  f  *  ^^^-y  sin  (x3  -  xy)  dx.  (8 

We  assign  the  value  0  to  the  integrand,  for  a;  =  0. 

To  show  that  this  integral  is  uniformly  convergent  in  any  33  =  (a/3))  let  us  use 
the  method  of  662.     If  we  set 

M  =  —  ,         dv  =  (3x^  —  y)  sin  (x^  —  xy) ; 
Sx 

/"»        _  r     cos(a;3  —  x?/)n°°      r'°cos(x^  —  xy) 

Jx  ~\_  3  X  J.c        Jx  Sx^ 

cos  (x^  —  xy)      f"^  cos  (ic^  —  xy) 


f 


8  X  J-r  3  x^ 


dx. 


Here  both  terms  are  uniformly  evanescent  in  33  by  660,  2.    Hence  8)  is  uniformly 
convergent  in  33. 

7.  I    cos  (x^  —  xy)dx.  (9 

We  can  write 

'3x^  —  y  „  y  C^  COS  (x^  —  x?/) 


i    cos  (x*  —  xy)dx  —  \     —     .^     cos  (x^  —  x?/)c?x  +  o  l 


X' 


dx. 


The  second  integral  on  the  right  is  uniformly  evanescent  by  660,  2.     The  first 
integral  is  also  uniformly  evanescent.     For  integrating  by  parts, 

J'=°  3  x2  —  w        ^  „         ^  ^           sin  (x^  —  xw)      2  f »  sin  (3  x^  —  xy)  , 
,    ^^'C0S(XB-X,)dx  = W^  +  3i  XB  ^^- 

Here  both  terms  on  the  right  are  obviously  uniformly  evanescent. 

664.    lieif(xi/y  be  regular  in  R=  (aaoa/3),  yS  finite  or  infinite. 
Let  ^00 

converge  uniformly  in  ^,  except  possibly  at  «j,  •••  a^.  To  establish 
the  uniform  convergence  of  «/ throughout  ^,  we  have  only  to  show 
that  J  is  uniformly  convergent  in  each  of  the  little  intervals 

That  is,  we  have  only  to  show  that  for  each  e  >  0,  and  for  some 
S  >  0,  there  exists  a  b^  such  that 

for  any  i/  in  ^^  and  every  b'  >b^,  k  =  1,  2  "•  m. 


UNIFORM   CONVERGENCE  473 

665.    Examples. 


1-  j^  ^-smysmxy^^ 


is  uniformly  convergent  in  53  =  (0,  oo). 

For,  in  the  first  place,  the  integrand  f{xy)  is  continuous  in  i2  =  (0,  oo,  0,  oo)  if 
we  set 

/(O,  y)  =y  sin  y. 

We  have  therefore  only  to  consider  the  singular  integral  relative  to  a;  =  oo,  in  the 
intervals  i8i  =(0,  5),  $82  =  (5,  00).     Now  as  in  663,  2,  we  have  for  y>0, 


-^  p  sin  y  sin  xy  ^^  _  sin  y  cos  xy  _  ^^^^      pcos  xy 

Jx  X  XV  J.C        X^ll 


The  reasoning  of  663,  2  shows  that  iTis  uniformly  evanescent  in  352. 
As  to  S81,  we  note  first  that  ^  =  0  f or  y  =  0.    Also  that 

sin  y      1    ,     ;         I    / 1  ^ 

— ^=1  +  7;',      h  K'7) 
y 

17  being  as  small  as  we  choose,  if  5  is  taken  small  enough. 
Hence  for  any  ?/  in  S3i, 

X  Jjo     X^ 

which  shows  that  ^is  uniformly  evanescent  in  S3i. 


r^Hi^dx,    x>o  (1 

Jo     weAi 


y& 

is  uniformly  convergent  in  33  =  (0,  /3). 

For,  the  integrand /(a;?/)  is  continuous  in  jB  =  (0  co  0  /3),  if  we  set 

Let  us  consider  therefore  the  singular  integral  relative  to  x=oo.  We  set  35i  =  (0,  5), 
332  =  (5,  oo).     Obviously  1)  is  uniformly  convergent  in  332. 
To  show  the  same  for  33i,  we  note  that 

sin  xy  =  xy  +  TX^y^,        1  t |<  1, 

by  the  Law  of  the  Mean.    Hence 

|/(«y)(<^  +  ^',        m(0,  Qo,0,S). 

Thus,  by  659,  the  integral  1)  is  uniformly  convergent  in  Si. 


474  INFINITE   INTERVAL   OF   INTEGRATION 

Continuity 

666.    1.  Let  f{xy)  he  regular  in  R  —  (acC)  a/3),  ^finite  or  infinite,, 
except  on  the  lines  x  =  a^  ■■•  x  =  a^. 

±.  Let  /^« 

J=jJ(xy^dx 

he  uniformly  convergent  in  ^. 

2°.   Let  Vim  f(^xy)  =  (f)(x^,,         rj  finite  or  infinite 

uniformly  in  any  (a,  5),  except  possihly  on  the  lines  x  =  a^  ••• 
Then 

y=]im   )  f{x,  y^dx         exists.  (1 

3°.   Let  4){x}  he  integrahle  in  any  (a,  J). 
Then 

\\xiiJ=\\m   I  f{xy^dx=  I   ^(x)dx.  (2 

By  virtue  of  616  we  may  assume  that/  is  regular  in  jB,  and  that 
f==<f>  uniformly  in  any  (a,  6). 

To  fix  the  ideas  let  rj  =  ao. 

We  show  first  that  j  exists;  i.e.  for  each  e>0  there  exists  a  7 
such  that 

D=  f  \f(x,y'}-f(x,y")ldx 


is  numerically  <  e  for  any  pair  of  numbers  y',  y"  >y. 
Now 

i>=  rV(^/)  -fCxy"}\dx-hrf(xy'yx-  ff(xy")dx 

=  i)l  +  1)2  +  2)3. 

By  1°,  there  exists  a  h  such  that 

lAJ,   |i)3|<e/4  (3 

for  any  ?/',  y  in  ^. 

By  2°,  we  can  take  7  such  that  for  any  x  in  (a,  5) 

l/(^y)-K^)l<77^^^        y>7. 


CONTINUITY  475 

Hence 


l/(-,y)-/(.,y")i<2(jr^. 

for  any  ?/',  y"  >7,  and  x  in  (a,  h'). 

Thus  I  T^  I        /o  /--( 

|i)i|<e/2.  (4 

From  3),  4)  we  have  ,  -r,, 

|-^l<e- 

We  rzea^f  s^ot^;  that  2)  holds ;  i.e.  for  each  e>0  there  exists  a  5^, 
such  that  I  ^j        I 

\j  —  I  <^c?3;  <c  (5 

for  any  h>hQ. 

From  1),  there  exists  a  7  such  that 

j=jj{xy^dx  +  e\         |e'|<|  (6 

for  any  ?/  >  7. 

From  1°,  there  exists  a  h^  such  that 

jy(ixy^dx=jjixy^dx  +  e'\  \e"\<±        (7 

for  any  h  >  b^,  and  any  y  m  ^. 

From  2°,  we  can  take  7  large  enough  so  that  also 

f(x,y)  =  ct>{x}  +  g(xy'),         \g\<-^L^      (8 

for  any  x  in  (a,  6),  and  any  y  >  7. 
Hence,  by  3°,  ., 

jy(a:z/y:r=J^(/,(:r)(i:r  +  e'",         l6'"I<|  (9 

for  an}^  h  >  J^,  and  any  ?/  >  7. 
From  6),  7),  8),  9)  we  have 


for  any  b  >  5q.     But  this  is  5). 

2.  The  reader  should  note  that  the  lines  aj=  a^,  •••  on  which  the 
uniform  convergence  of  f(xy')  to  <f)(x')  breaks  down,  are  according 
to  617,  3  singular  lines,  whether  f(xy')  has  points  of  infinite  dis- 
continuity on  them  or  not.     If  the  integral  J  is  to  be  uniformly 


476  INFINITE   INTERVAL   OF   INTEGRATION 

convergent  in  ^,  the  singular  integrals  relative  to  all  these  lines 
must  be  uniformly  evanescent. 

667.    Example.     As  we  shall  show  in  675, 

J  =  \ —  sin  \x  dx  =  arc  tg  ^,       X  ::^  0. 

Jo  X  X 

The  application  of  666  gives  .     ,„        x  ^  n 

limJ=  r"?Hi2^(^x=      0,  X  =  0.  (1 

y=X  J^  X  \ 

[  -  7r/2,  X<0. 

That  the  conditions  of  Theorem  666  are  fulfilled  is  easily  seen.  For,  defining  the 
integrand /(x?/)  of  </as  in  663,  3,  it  is  continuous  in  R  =(0,  x>,  0,  oo).  We  also  saw 
that  the  singular  integral  relative  to  sc  =  oo  is  uniformly  evanescent  in  i8=  (0,  oo). 

Now  .    , 

T„  ^/        N      sin  Xx 
hm  /(a;,  y)  = 

9=00  X 

uniformly  in  21  =(0,  oo)  except  at  x  =  0. 

The  line  x  =  0  is  therefore  a  singular  line  by  617,  3.     But 


\    sm  Xx  c?x  <    \    dx  <  e,         0  <  a  <  5 

J«  X  I       IJo        X  I 


if  5  is  taken  sufficiently  small.     This  singular  integral  is  therefore  uniformly  eva- 
nescent.    Hence  J  is  uniformly  convergent  in  58.     Thus  all  the  conditions  of  666 
are  satisfied. 
From  1)  we  may  deduce  the  following  relations  : 


psinjxxcos^^^^  (2 

Jo  X 

0<a</3. 


X 


COS  ax  sm  )3x  j    _  w;  /o 

'0  X  ~2'  ^ 


which  may  be  comtjined  in  a  single  formula 


r 


sin  Xx  cos  /ua;  ^    _  [  0,       0  <  X  <  /n. 


/o  X  [7r/2,  0<Ai<X 

In  fact,  since  a  +  /3  >  0,  a  —  ^  <  0,  we  have  from  1), 


X  2'  '^ 


sin(a-^)x^^^_x  .g 

'0x2  ^ 

But 

sin  (a  +  /3)x  +  sin  («  —  ;3)x  =  2  sin  ax  cos  /3x,  (7 

sin  (a  +  /3)x  —  sin  («  —  ^)x  =  2  cos  ax  sin  /3x.  (8 
Adding  and  subtracting  5),  6)  and  using  7),  8),  we  get  2),  3). 


CONTINUITY  477 

668.    That  the  relation 

Km  j  f(x,y')dx=  )     \\mf(x,y^dx  (1 

is  not  always  true  is  shown  by  the  following  example : 
From 

f  e-^  sin  X  dx = -  i::!l55^^+x!iE^, 

J  1+  W2 


we  have  for  ?/  >  0,  /♦^o  ^jj^  ^  ^ 


r?i^dx  =  -J— .  (2 

Jo        (xy  1  +  m2 


gxy  1+2/' 

y=o  Jo      e^y 


On  the  other  hand, 

P2?lim515^dx=  (""sinxdx 
Jo        v=o    e="J'  Jo 


70        3,=o     e=^J' 
does  not  even  exist. 

Thus  the  relation  1)  does  not  hold 


669.    1.   Let  f{xy)  he  regular  in  M  =  (^acca^^  except  possibly  on 
the  lines  x=  a^^  •••  x=  a^. 

Let  f(xy^  he  a  uniformly  continuous  function  of  y  in  ®  except 
possihly  on  the  lines  x  =  a^^  •••     Let 


%/  a 


he  uniformly  convergent  in  ^. 
Then  J  is  continuous  in  ^. 

For,  f{x,  y  -{-  h')  converges  uniformly  to  /(a;,  y),  h  =  0  in  ^ 
except  on  the  lines  x  =  a^,  •  •  •  We  have,  therefore,  only  to  apply 
666. 

2.  Letf(xy^  he  in  general  regular  ivith  respect  to  x  in  R=  (^acca^^. 
Let  f  he  in  general  a  semi-uniformly  continuous  function  of  y  in  ^. 
Let 

J'iy')=  )  Kxy^dx 

he  uniformly  convergent  in  ^. 

Then  'lis  limited  in  ^,  and  in  general  a  continuous  function  of  y. 


478  INFINITE   INTERVAL   OF   INTEGRATION 

For,  we  can  take  h  so  large  that 


<€ 


for  any  y  in  >&.     On  the  other  hand, 

is  limited  in  ^  by  618,  1.     Hence  J  is  limited  in  iB.     That  J  is 
in  general  continuous  in  ^  follows  from  1. 

3.   In  this  connection  let  us  note  the  following  theorem  whose 
demonstration  is  obvious. 

Let  f{xy')  he  regular  in   R  =  {ayDa/3'),  /3  finite  or  infinite,  except 
on  the  lines  x  =  a^  ■••  ;  y  =  a^^  ■•■     Let 

1  K^y~)dy 

he  uniformly  convergent  in  any  (a,  5)  except  at  a^  a^  •••      Then  the 
points  of  infinite  discontinuity  of 

fixy^dy,        y  in  «. 
must  lie  on  the  lines  x^  ay,  ••• 

670.    1.  Letfixy')  he  regular  in  R=  (aooayS),  except  on  x^a^-"  ; 

y  =  H- 

1°.    Let 


Sj'y 


converge  uniformly  in  any  (a^  b)  except  at  x—  a-^ 

2°.    Let  ^:c       ^y 

4>(y)=  )     dx\  fdy 

converge  uniformly  in  '^. 
Then  cf)  is  continuous  in  ^. 

This  is  a  direct  application  of  666,  1,  where 
g(xy)  =  I  f(xy)dy 
takes  the  place  of/ in  that  theorem. 


rNTEGRATION    AND   mTERSlON  479 

In  fact,  by  669,  3,  g{xy)  has  no  points  of  infinite  discontinuity 
except  on  2;  =  ai  •••;  and  is  therefore  by  2°,  regular  in  R  except 
on  these  lines. 

Also  g{x^  J/  4-  ^)  converges  uniformly  in  any  (a,  6)  to  g{x^  if) 
as  ^=  0,  except  at  rc  =  a^  ••• ;   since 


V 

is  uniformly  evanescent  in  (a,  6)  by  1°. 
Thus  applying  'o^^^  1,  we  have 

lim  0(?/  +  li)  =  lim  |    g(x.  y  +  li)dx  =  I    lim  g{x,  y  +  Ti)dx 

=  J[   dxjydy=^<^iy^. 

That  is,  <^(«/)  is  continuous  at  ?/. 

2.   As  a  corollary  of  1  we  have  : 

Letf(xy')  he  in  gerieral  regular  tvith  respect  to  xin  R=  («ooa/3). 

converge  uniformly  in  Sd-     Tlien  (f)  is  continuous  in  SS' 

Integration  and  Inversion 

671.  1.  Let  f(xy')  he  in  general  regular  with  respect  to  x  in 
R=  (^aooa^).  Let  f  he  in  general  a  semi-uniformly  continuous 
function  of  y  in  SQ.     Let 


i(y)=  j  fC^y~)<i^ 


be  uniformly  convergent  in  ^. 
Then  Us  mtegrahle  in  ^. 

This  follows  at  once  from  500  and  669,  2. 

2.  As  a  corollary  of  1  we  have : 

Letf(xy')  be  simply  irregular  ivith  respect  to  x  in  M=  (aQoa/3). 
Let  J  be  uniformly  convergent  in  ^.      Then  J  is  integrable  in  ^. 


480  INFINITE   INTERVAL   OF   INTEGRATION 

672.    1.  Let  f{xy)  he  in  general  regular  in  i2=(aooa^),  wiil{ 
respect  to  x. 

1°.    Let  ^„ 

I    fdx  (1 

he  uniformly  convergent^  and  integrahle  in  Sd. 
2°.  Let 


'a  *^a 

admit  inversion  in  ^. 


dy  I  fdx^         h  arhitrarily  large^ 


Then 


We  set 


Jdy  I    fdx  =1     dx  \  fdy^         in  ^. 

rfdx=f\r.  (2 


Since  1)  is  uniformly  convergent  in  ^,  there  exists  for  each 
<•  >  0,  a  5q  such  that 

\rfdx<-^,  (3 

Wh  H  —  ft 


/3-a 


for  any  y  in  ^,  and  every  h  >  h^. 
Thus  2),  3)  give  for  any  y  in  ^, 


j    dy  \    fdx-  j   dy  \  fdx\<  I    ^  ^  <6.  (4 

But  by  2°, 

Hence 

J ^2/  /*=°         /'ft   /*y| 
J     -J   J    k^'         ^>^o>  (5 

,      a     ^a  ^a     ^  a.     \ 

which  proves  the  theorem.  • 

2.  Let  f(xy')  he  simply  irregular  with  respect  to  x  in  R=  (aooa/3)< 
Let 

jjdx 

he  uniformly  convergent  in  ^. 


mTEGRATIOX   AND   INVERSION  481 

Then  ^^ 

J    dyj   fdx,  J    dxj^  fdy  (6 

are  convergent  and  equal. 

For,  bj  671,  2,  the  integral  on  the  left  of  6)  exists. 
Moreover,  condition  2°  of  1  is  fulfilled,  by  622,  2. 

3.   As  corollary  of  1,  we  have : 

Let  f(xy^  he  in  general  regular  in  R  =  (aQoa/3)  with  respect  to  x. 
Let 

Cfdx 

he  uniformly  convergent^  and  integrahle  in  ^. 
Let 

I    dy  \  fdx^        h  arbitrarily  large, 

admit  inversion  in  ^. 

Then  ^^      ^^ 

I    dx\  f{xy')dy 

is  uniformly  convergent  in  ^. 

This  follows  at  once  from  5),  since  this  inequality  holds  for 
any  y  in  ^. 

4.   From  the  relation  4)  we  have  also  the  following  corollary, 
setting  y  =  (3. 

Let  f{xy^  he  in  general  regular  in  R  =  (aoDa/3)  with  respect  to  x. 
Let 

Cfdx 

he  uniformly  convergent.,  and  integrahle  in  ^.     Let 

Jdy  I  fdx.,         h  arhitrarily  large, 
admit  inversion.      Then 

lim  }    dyl  fdx  =  \    dy  \    fdx. 

6=00   *^a  ^a  »'a  *^a 


482  INFINITE   INTERVAL  OF  INTEGRATION 

5.  As  a  special  case  of  4  we  have,  622,  2  and  671,  2 : 

Let  f(xy)  he  simply  irregular  with  respect  to  x  in  R  =  (a<X)a^'). 
Let 

)    fdx 

he  uniformly  convergent  in  ^.      Then 

lim  I    dy  I  fdx  =1    dy  \    fdx. 

673.    Let  f(xy^  he  regular  in  It  =  (^a<x>a^'),  except  on  the  lines^ 
x^ay,    ■■  x  =  a^\    y  =  a-^,---y  =  a,. 

I    dy  I    fdx 

he  coiivergent^  and  admit  inversion  in  any  interval  (\,  fi),  which  does 
not  embrace  a^  •••  a^. 

J    dxj  fdy 

he  a  continuous  function  of  y  in  ^. 
Let 

^=  1    dy  j    fdx,         L=  \    dx)  fdy. 

Then  K  is  convergent,  and  K=  L. 

Foi-  simplicity,  let  3/  =  7  be  the  only  singular  y-line  ;  a  <  7  <  yS. 
Then  by  definition, 

dy\    fdx=\xm\     I     +limj     j    .  (1 

a  ^'a  u=y  ^a.    *^ a  v=y  ^^ *'    *'« 

a<u<ry,         <y<v<^. 

oimiiariy,  -,^  ^od       ^a^  ^o       p'^  r>^       /»=<=  /»» 

eince  i  is  by  2°,  convergent. 


INTEGRATION    AND   INVERSION  483 

Now  by  2°, 

Hence  from  2),  4), 

lim  J"  r=rr;  (6 

also  from  3),  5) 

lim  rr==rr-rr=rr.         o 

Hence  from  1),  6),  7)  we  have 

*/(i    tJa  J  a     *^a  ^a     ^y  *^a     *^a 

674.    1.   Let  f(xy)  he  simply  regular  in  Ii  =  (acca^')^  except  on 
the  lines  x  =  a-^,  ■••  y  =  a.^... 

1°.  Let 

)  K^y^dx 

converge  uniformly  in  ®,  except  on  y  =  a^  ••• 
2°.  Let 

converge  uniformly  in  any  (a,  5),  except  on  x—a^  ••• 

3°.  Let 

J     ^aj  f(xy~)dy 

converge  uniformly  in  ^. 

-Br=J    (^yj    fdx,         L=J    dxj  fdy. 

Then  K  is  convergent  and  K=  L. 

This  follows  from  673.     For,  in  the  first  place,  condition  1°  of 
673  is  satisfied,  by  672,  2. 

Secondly,  condition  2°  of  673  is  fulfilled,  by  670,  1. 


484  INFINITE  INTERVAL  OF  INTEGRATION 

2.   As  a  corollary  of  1,  we  have : 

Let  fixy')  he  simply  irregular  with  respect  to  x  in  ^  =  (aQoa/3). 
Let  ^00 

I    f{xy~)dx 

converge  in  general  uniformly  in  ^.     Let 

j  <^^j^A^y)dy 

converge  uniformly  in  ®.      Then 

J    <^yj  fdx=J    dxj  fdy. 
675.   For  y>0,  we  have  from  668,  2), 

r^^^^^ax^^—.  (1 

The  integral  on  the  left  does  not  exist  for  j/  =  0.    Let  us  therefore  set 

sill  \x 

=  0,  2/  =  0. 

Then  integrating  1),  from  0  to  y  we  get 

|>j;>x  =  j;^^d,  =  arctg^,         X:^0.  (2 

We  may  invert  the  order  of  integration  in  2)  by  674,  2.  For,  /  is  continuous  in 
i?  =  (OQoOy),  except  on  the  line  ?/ =  0,  and  limited  in  J2.  It  is  therefore  simply 
irregular  with  respect  to  %.  The  integral  1)  obviously  converges  uniformly  in  33 
except  at  2/  =  0.     The  integral 

J^dxJ^/fZ^  =  j^   —^sinXx^x  (3 

is  uniformly  convergent  in  33  by  663,  3.     Hence  the  integrals  on  the  left  in  2),  3) 
are  equal,  and 

/•«  1  _  g-xy     \  y 

t     sm  Xx  dx  —  arc  tg  - ,        X  ^  0. 

Jo         X  ^  X 


676.    We  saw  in  667  that  ^^  ^.^  ^^,  f  0,         y  =  0, 

li 


f  =°  sm  xw  ,  "  ,, 

Jo        X  !  -,        2/>0. 

Hence,  integrating  between  0  and  1 ,  we  get 


INTEGRATION   AND   INVERSION  485 

"We  can  invert  the  order  of  integration  by  674,  2.  For,  in  the  first  place,  the 
integrand,  not  being  defined  in  2),  we  can  make  it  continuous  in  B  =  (0x01),  giving 
it  the  value  y  at  the  points  (0,  y).  Secondly,  the  integral  1)  is  uniformly  conver- 
gent in  33,  except  at  y  =  0,  by  663,  2.     Finally, 

p^    r.sinxy  pl-cosxy^^ 

Jo         Jo         X  ^       Jo  X2 

is  uniformly  convergent  in  53,  since 

1 1  —  cos  xy  I      2 

I        ^        |  — x2' 

We  can  therefore  inveit  in  2),  which  gives 

IT      f"  cZx  fi  .          ,        f "  1  —  cos  X  J 
-=l     — I    sm  xy dy  =  \     ax 

2     Jo    X  Jo  "   "     Jo         x2 

^2psin2x/2^^^psin^j£d«  .3 


setting  X  =  2  M. 

Thus  3)  gives 


/*°°  sin^x  dx  _v  ,. 

Jo        x2      ~  2 '  ^ 


677.   That  the  order  of  integration  can  not  always  be  inverted  is  shown  by  the 
following  examples. 

Ex.  1.   Let  us  consider 

I    dx  j    co&xydy=\     —  dx  (1 

TT 

~  2' 
The  integral  obtained  by  inverting  the  order  of  integration,  viz., 

i    dy  i     cos  xy  dx 
does  not  exist,  since 

1     cos  xy  dx 

does  not.     Inversion  in  the  order  of  integration  in  1)  is  therefore  not  permissible. 

Ex.  2.    Let 

— -= =  0(m),        u  =  xy. 

1  +  xV     1  +  M* 
Then 

Let 

/(x,2/)=0'OO=^f -if- 
y  dx      X  dy 


486  INFINITE    INTERVAL    OF   INTEGRATION 

AVe  have  also 

where  lA  («)  =  arc  tg  u^. 

Thus 

and  hence 

^  0(m) ^  3f 
a;         5a; 
Hence 

=irg-=.-[^«]:=!- 

On  the  other  hand 

Thus  it  is  not  permissible  to  invert  the  order  of  integration  in 

]-;..!;/(.,  i/)^.=f  ^x|;|  ^ .,.  (2 

678.    1.   Letf{xy^  he  regular  in  _B  =  (aQoaQo),  except  on  the  lines 
x  =  a-^,  •••;  y  =  ai,  ••• 

1°.  Let  ^«       ^y 

5e  uniformly/  convergent  in  Sd- 

he  uniformly  convergent  in  any  (a,  5),  except  at  a^,  a^-,  •••;   awe?  inte- 
grahle  in  (a,  6).     Then  ^^       ^^ 

I     t(?a:  I    fdy         exists^ 

lim  I     t?a;  I  /(?y  =  I     (^j:  1    fdy.  (1 

This  is  a  direct  application  of  (SQQ,  1 ;  the  function 

fixy^dy 

a 

taking  the  place  oi  f(xy^  in  that  theorem.     For,  in  the  first  place, 
g  has  no  points  of  infinite  discontinuity,  except  on  the  lines  x  =  a^ 


INTEGRATION    AND   INVERSION  487 

•••,  by  669,  3.     Moreover,  g{xy)  is  integrable  in  31,  by  1°.     Hence 
gi-ry^  is  regular  in  i?,  except  on  the  lines  x  =  a^,  ••• 
Secondly, 

I    g{xy)dx 

is  uniformly  convergent  in  ^,  by  1°. 
Finally 

lim  gixy')  =  I    fdy  =  ^(x), 

uniformly  in  any  (a,  6),  except  on  the  lines  x  =  a^^  •••;  moreover 
(^  is  integrable  in  (a,  5).  Thus  all  the  conditions  of  666,  1  are 
satisfied,  and  the  present  theorem  is  established. 

2,  As  a  corollary  of  1  we  have : 

Letf(xy^  he  simply  irregular  with  respect  to  y  in  R  =(aQt)aQo). 
Let 

I    dx\  fdy 

be  uniformly  convergent  in  -53.     Let 

be  uniformly  convergent  in  any  (a,  6).      Then 

dx  I  /ii/  =  )    dx\  fdy.   , 

For  2)  is  integrable  in  any  (a,  6),  by  671,  2 ;  on  interchanging 
a;  and  y  in  that  theorem. 

679.    If  the  conditions  of  678  are  not  satisfied,  the  relation 

lim  \    dx\   fdy  =  \    dx  I  fdy  (1 

may  be  untrue. 

Consider,  for  example, 

J=\    dxi  cosxydy,        a>0. 

J»         X 


488  INFINITE    INTERVAL   OF   INTEGRATION 

Here  limj"=0,        by  663,  Ex.  2. 

On  the  other  hand,  the  integral 

I    da;  t    cos  xy  dy 

Ja  Jo. 

does  not  even  exist,  since 

I    cos  %y  dy 

does  not.    Thus  the  relation  1)  in  this  case  is  not  true. 

680.    1.  Let  f(xy^  he  simply  regular  in  i2  =  (aQoaao),  except  on 
the  lines  x=  a-^,  •••',  y  —  (f-y,  ••• 
1°;  Let  ^^ 

I  fdx 

he  uniformly  convergent  in  any  (a,  /3)  except  at  «j,  ♦•• 

jjdy 

he  uniformly  convergent  in  any  (a,  5)  except  at  a-^  •••;  moreover  let 
it  he  integrahle  in  (a,  J). 
.  3°.  Let  ^a=      ^y 

j   dx  j  fdy 

he  uniformly  convergent  in  Sd- 

Then 

■'■''''-'If'  ptn  ^CXJ  ^00  ^00 

I   dy  j  fdx,         I    dx  I  fdy 

are  convergent  and  equal. 
For,  by  674, 1, 

)   dy)  fdx  =  i   dx  i  fdy. 

.  But  by  678,  1,  we  may  pass  to  the  limit  /3  =  oo,  which  proves 
the  theorem. 

2.  LetfQcy^  he  simply  regular  with  respect  to  y  in  R  =  (aoc  aco), 
except  on  the  lines  y=  a^  ••• 
Let 


INTEGRATION    AND   INVERSION  480 

he  uniformly  convergent  in  any  («,  /S)  except  at  Oj,  ••• 
Let  ^^ 

he  uniformly  convergent  in  any  (a,  6). 

Let 

I   dx  I  fdy 

he  uniformly  convergent  in  ^. 
Then 

are  convergent,  and  equal 

This  follows  as  in  1,  by  674,  1,  and  678,  2. 

3.  Let  f(x,  2/)>0  ^^  simply  regular  in  Ii  =  (^accacc^  except  on 
the  lines  a;  =  <Zj,  •  •  • ;  y  =  a^,  •  •  • 

Let  ^« 

Je  uniformly  convergent  in  any  (a,  /3)  except  at  aj,  ••• 

he  u7iiformly  convergent  in  any  (a,  J)  except  at  a^^  "- 

fte  convergent.      Then 

K=£dyjjdx 

is  convergent,  a7id  K=  L. 

This  is  a  corollary  of  1.  Yov,  condition  2°  is  satisfied  since  L 
exists.  That  condition  3°  is  fulfilled  follows  from  the  fact  that 
the  singular  integrals  of 

are  <  the  corresponding  integrals  of  L  since  /^  0. 


490  INFINITE   INTERVAL   OF   INTEGRATION 

681.    1    We  saw  in  667,  1)  that 

/*"  sin  xy  cos  Xx  ,         f  0»       ^  <  ^        .       =  ^ 

Multiply  by  e-Mw,  and  integrate,  /a  >  0.     Then 

Jo     "  Jo  xet^y  JO  Jo      Ja  ^ 


2  ixe^i^ 
"We  can  invert  the  order  of  integration  in  1)  by  680,  2.     For,  in  the  first  place 

. ,     .      sin  xy  cos  Xx 

f(xy)  = 

is  simply  regular  in  Jr!  =  (OcoOoo),  if  we  set 

Secondly, 

C  ^-,  r°^  sin  xw  cos  Xx  J 

\  fdx  =  e-y-y  \ dx 

Jo-'  Jo  X 

is  uniformly  convergent  in  39  =  (0,  »)  except  for  ?/  =  X  by  663,  Ex.  4. 
Thirdly, 

Cfdy  =  cos  Xx  P  ^'"  ^y  dy         for  x  >  0 
Jo*^   ^  Jo     xei-y 


\     —dy        for  X  =  0 

Jo    eMy    ^ 


is  uniformly  convergent  in  any  (0,  6)  by  66.5,  Ex.  2. 
Finally, 

r=  Cdx  {jdy  =  Cdx  (^J^^.^im^ay 
Jo       Jo-'  ^      Jo       Jo         xei^y  " 

is  uniformly  convergent  in  33. 
For, 


Hence 


fysinxy       _r       _     /xsinxy  +  xcosxy"]!' 
Jo    et^y     2/-|_-e  >^y  ^12^7^2         Jo' 

f^cosXx  ..  „„  .,  r^'sinxw     cos  Xx   , 

^=Jo   ^^^,(.'^-e->^^'^o^xy)dx-,.e-.y)^   __l.__d^ 


mTEGRATION    AND   INVERSION  491 


Fi,  Y2  are  uniformly  convergent  in  35  by  659.     For, 


cos  Xx  ,,  .  2 

I  sin  xy  cos  Xx 


I     X      ;u'-2  +  x"-^| — M^  +  k' 

Thus  all  the  conditions  of  680,  2  are  satisfied. 
Inverting  therefore  in  1),  we  get 


fl^  +  X- 
Comparing  with  1),  we  get 


^      C  '  cos  Xx  ,     C    sm  xy  , 
K  =  I     dx  \     =-  dv 

Jo        X  Jo      e>^» 

f  cos  Xx  ,    r  fjL  Bin  xy  +  X  cos  xtf~| 

Jo       X  L  /x2  +  x2  J 


/i^  -f  x'^  Jo 

cos  Xx   , 


f       cos  Xx     ,  W  ^    =  rt  ^    A  /-o 

\     -r, 5  dx  = ^-        X  >  0,  /A  >  0.  (2 

2.  Let  us  integrate  2)  with  respect  to  X.     We  get 

fA       r-  cos  Xx    ,        T   f  A     ^    ,^ 

=  2^(1 -e-V).  (3 

We  can  invert  the  order  of  integration  in  the  integral  on  the  left,  by  674,  2. 

For, 

r'"  cos  Xx 
Jo    /i^ 


M^  +  X^ 

is  uniformly  convergent  by  659,  since 


dx 


cos  Xx  I  1 


\fj:-^  +  x;^i  —  ii^  +  z^ 
la  the  second  place, 


C       dx      C^       ^     7^       f        smXx      , 
I     -3 5  I    cos  \xd\=  \    —r-i, sv  dx 

Jo     /i2  +  ^2  Jo  Jo     x(ai2  +  x2) 


converges  uniformly  in  any  interval  (0,  /3)  by  661,  2  and  663,  Ex.  2. 
Inverting  therefore  in  3) ,  we  get 


f         sin  XW  ,  TT     ,,  -  ^    n  =  rw 

jo  ^0^^"  =  V^'-'""'^'         ^>0,2/>0. 


492  INFINITE   INTERVAL   OF   INTEGRATION 

682.    Let  us  evaluate  ^co  ^,,  * 

J=  j    %  (1 

Jo    g«2'  '^ 

which  is  convergent  by  635,  3. 

We  change  the  variable,  setting 

u-xy,        y>0.  (2 

Then  ^„    ^ 

j_  C   y  dx^ 

Jo     gff'' 

Multiplying  by  e-»*  and  integrating,  v?e  get 

'  dy    r°"  7  r*  y  ^* 


jC%=CclyC 
Ja  e»       Ja       Jo 


^=(1+1=) 


This  relation  is  true  for  any  a  >  0,  by  2). 

Passing  to  the  limit  a  =  0,  we  have,  since  the  limits  exist, 

'  dy      j.^      ("^  ,    C"    y  dx 


^    c^  dy     -TO     C^  n   C"  y  dx 

-"•Jo   J  =  ^=io^^^jo   ifc^)-  (3 


"We  may  invert  the  order  of  integration  in  the  integral  on  the  right  by  680,  3. 
For,  in  the  first  place,  the  integrand  is  regular  and  continuous  in  B  =  (OcoOco). 

S^^^^^^y'  p    ydx 

Jo    e»'(i+^'> 

is  uniformly  convergent  in  any  (a,  /3) ,  a  >  0  by  659,  since 

y      ^     P 

Thirdly,  ^ 

ydy 


r 


is  uniformly  convergent  in  21  =  (Oco)  by  659,  since 

y  _^y_, 

_2^  5^       ..2* 


Finally, 

Jo       Jo   ej^(i+x==)      Jo       L     2(1  +  x''')Jo 
dx        IT 


2J0 1 


+  a;2     4 

is  convergent.    Thus  all  the  conditions  of  680,  3  being  fulfilled,  we  can  invert  in  3), 

which  gives 

J^  =  L  =  ir/4. 

Hence 

J=±-y/^/2. 

Here  we  must  take  the  positive  sign,  since  the  integral  1)  is  positive  by  649,  3. 

ITpiicp.  finally,  .- 

r°"  dx  _  V  X  ^  J 

Jo   I^~~2~* 


DIFFERENTIATION  493 

Differentiation 

683.    1.   Let  f(xy')^  f\j{xy^  he  in  general  regular  with  respect  to 
X,  y  in  11=  (aye  «/3 ) . 

1°.   Fo)'  each  x  in  91  let  f  he  continuous  in  y,  while  f'y  is  in  general 
eo7iti7iuous  in  y. 

2°.   For  each  y  in  ^,  let 

I  dx  I  f'ydy^         h  arhitrarily  large^ 
admit  inversion.      Then 

ay  *^a  dy  &=«  »^a      *^a 

provided  the  derivative  on  either  side  exists. 
For,  by  605, 


Hence 


and  therefore 


J  fAy  =f(x,  y')  -fix,  «). 

1  fdx  =1   dx  \  fydy  -\-   I  /(a;,  a)dx 

X'j       rb  r-b 

dy  I  f'ydx+  I  f{x,  ft)dx; 
a.  *^a  *^a 

I   fdx=\\m  I   dy  I  f'ydx+  I  /(a;,  «)(fa 


Differentiating,  we  get  1),  since  the  last  term  on  the  right  is  a 
constant. 

2.   As  corollary  of  1  we  have  : 

Letf{xy)  he  regular  in  i2  =  (fflco«/3)  except  on  the  lines  a;  =  a^,  ••• 
and  continuous  with  respect  to  y  for  each  x  in  21. 

Let  f'y  he  regular  in  R  except  on  the  lines  a;  =  a^,  •••  and  uniformly 
continuous  in  y,  except  on  these  liiies. 

Let 

I  fydx 

he  uniformly  convergent  in  ^. 
Then  ^ 

-3-  1  .f(xy^dx  =  I  fydx. 


494  INFINITE   INTERVAL   OF   INTEGRATION 

For,  condition  2°  of  1  is  fulfilled  by  622,  2. 
Hence  by  1), 

3-  I  f(x^}dx  =  —  Yim  I   di/  I  fi,dx 

=  -T-  f'^^  f/^^a:,         by  672,  4,  and  671,  2, 
=  Cfydx,        by  669,  1. 
684.    1.  When  ^» 

\fydX 

is  not  convergent,  the  following  theorem  may  serve. 

Let  f{xy^  he  in  general  regular  in  R=  (^aooajS),  and  continuous 
with  respect  to  y  for  each  x  in  %. 

Let  fy  he  simply  irregular  with  respect  to  x  in  R. 

1°.  Let  , 

I  f'ydx^         h  arbitrarily  large, 

he  uniformly  convergent  in  ^. 
2°.  For  any  b,  let 

Jr»6  /»6 

fydx=  I  g(xy}dx-{-h(h,  y),         where 

3°.  g(xy^  is  simply  irregular  in  R  with  respect  to  x  and 

J/»ao 
I  gixy^dx 
a 

IS  uniformly  convergent  in  ^. 
4°. 


Then 


lim  I    h(b^  y^dy  =  0. 
^  =3-  )  fQ>^y^dx=  I  gQcy^dx.  Q 

dy^a  *^a 


DIFFERENTIATION  495 

2.   As  corollary  we  have  : 

Letf{xif)  he  regular  in  R=  Qaco  a^'),  and  continuous  with  respect 
to  y  for  each  x  in  %.     Letf'y  he  continuous  in  R.     For  any  h,  let 

Jr'b  /*b 

I  f'ydx=  J  g{xy)dx+  h(h,  y), 

where  g  is  regular  in  M,  and 

I  gi^y^dx 

is  uniformly  convergent  in  ^  :  also 

lim  I    A(5,  y^dy  =  0. 

Then  r,    ^^^  ^oo 

^Jy(^y)^^=J^  9(S^y')dx. 

For,  condition  1°  of  683  is  obviously  satisfied,  while  condition  2° 
is  fulfilled  by  672,  2.     Hence 

J"'  =  --  lim  I    dy  )  f'ydx. 

dy   6=«  *^a  '^a 

But  fy  fb  fy  fh  fy 

y  dyy  f'ydx  =  j  dyj^  gdx  +  J^  hdy,         by  2°. 

Hence  by  4°,  ^  ^y      ^b 

J'  =  —  lim   I   dy  I  gdx 

dy  ^a.  ^a 

=:^  rdxCgdx,         by  3% 
dy^a      ^a 

=  I  gdx,         which  is  1). 


EXAMPLES 


685.   1.  Let 

We  show  that 


j=r'-}i^dx.  (1 

Jo      are^ 

dJ^  rcosxy^^^        2/ arbitrary,  (2 

dy      Jo      e=^ 


using  683,  2.      For,  in  the    first   place,    the    integrand  f{xy)   is   continuous   in 
B  —  (Ox>a8),  if  we  set 

/(O,  y)  =  y. 


496  INFINITE  INTERVAL  OF   INTEGRATION 

Obviously  J  is  convergent  in  33,  by  635,  3. 

Secondly, 

f,  _  cos  xy 

e* 

is  continuous  in  B  ;  and 

pcosx,^^ 


is  uniformly  convergent  in  33,  by  660,  2.     Thus  683,  2  gives  2). 
By  means  of  2)  we  can  evaluate  1). 
For,  obviously, 

Jo       e^  1  +  2/2 

Hence  integrating  2),  we  get 

J  1  +  2/2 
Since  J"  =  0,  for  y  =  0,  we  have  C  =  0.     Hence 

r!!E^dx  =  arctgy.  (3 

Jo      xe^ 

2.  From  this  integral  we  can  also  show  that 

/•«  sin  X2/  T, 

Jo       X  2 

a  result  obtained  in  667,  by  the  aid  of  675.     For,  set 

x  =  ^*,        2/>0, 

y 

in  3),  we  get 

I     •  e  ydu  —  arc  tg  m.  k«» 

Jo       M 

We  now  apply  666,  1,  letting  y  =  cc. 
This  is  permissible,  since 

sin  u  _'i  .  sin  u  .,        , 

e  y— ,        uniformly 

u  u 

in  (0,  co)  except  for  u  =  0.     The  integrand  /(rt,  y)  is  continuous  in  i?  =  (Ocoaco),  if 
we  set 

/(O,  y)  =  1. 

The  only  singular  line  is  therefore  u  =  0. 

Obviously  the  singular  integral  for  this  line,  as  well  as  for  the  line  m  =  oo,  is 
uniformly  evanescent,  by  615  and  659. 

Hence  passing  to  the  limit,  y  =  oo  in  5),we  get 

Jo        M  2 

If  we  set  u  =  xy,  y>  0,  we  get  4) . 


686.   Let  ^'^  1  —  cos  xy 


diffp:rentiation  497 

^pl-cosxy^^  (1 

Jo         xe^  ^ 


Applying  683,  2,  we  get 

dJ_  r'^siii.xy  y 


aJ      C^siaxu  ^  y 

d^  =  }o   ^^^^  =  m^'       y  arbitrary.  (2 


In  fact,  the  integrand /(x,  y)  is  continuous  in  B  —  (Ocx)a/3),  if  we  set 

/(O,  2/)  =  0, 

while  J  is  convergent,  by  635,  3. 

Moreover 

_  sin  xy 
Jy-     ex 

is  continuous  in  i?,  and 

r"  sin  xy  , 
i    -d% 

Jo       &" 

is  uniformly  convergent  in  S,  by  660,  2.     This  establishes  2). 
As  in  685,  we  can  use  2)  to  evaluate  1). 
For,  integrating  2),  we  get 


J  1  +  V'     2 


+  y 

Here  C  =  0,  since  J  =  0  for  y  —  0,  by  1). 

Thus 

/'=°  1  —  cos  xy  ^       1 ,      ,,        „- 

687,    Let  us  evaluate  Fourier''s  Integral 

j^C-coB2xy^^ 

Jo  gxi'  ^ 

Using  683,  2,  we  get 

^=_2r  ^^i^2^^a;  =  ^.  (2 

-   dy  JO         e^2  "^ 

For,  the  integral  1)  is  convergent  by  635,  2  ;  while  the  integral  2)  is  uniformly 
convergent  in  any  («/3),  by  660,  2. 

In  2),  let  us  integrate  by  parts,  setting 

u  =  sin  2xy,        dv  =  —  2  xe-'*dx. 
Then 


K=  uv\    —  i    V 

Jo      Jo 


du 


=  —  2y\    g-*^  cos  2  x?/  c?x  =  —  2  yJ". 
This  in  2)  gives,  since  J"=jtO, 


^=~2ydy. 


498  INFINITE   INTEKVAL   OF   INTEGRATION 

Hence 

log  J=-y^+0.  (3 

To  determine  C,  take  y  —  0.     Then 

C  =  log-Vw/2,  (4 

by  682,  4). 

Hence  1),  3),  4)  give 

Jo        e^2  2 

688.   In  681,  2)  we  found 

rcosxy^^       _jn_  Q         y^a>0.  (1 

We  can  differentiate  under  the  integral  sign,  by  683,  2.     For,  denoting  the  inte- 
grand hy  f(xy),  we  have 

^ ,     ^  X  sin  xu 

which  is  continuous  in  E  =  (Ox>a(3). 
Also 

'  sin  xy       dx 


j;/,cte=-j'; 


1+^ 

is  uniformly  convergent  in  93,  by  661,  2. 
Hence,  differentiating  1),  we  get 


Jo      ;^2  +  a;2  2 


689.   In  682,  4),  let  us  replace  x  by  xa/a,  ?/>0.     We  get 


r  e-!'^VZa;  =  — 2/-5,        ?/>a>0.  (1 

We  can  differentiate  under  the  integral  sign,  by  683,  2,  getting 

Cx^e-y^^dx  =  —  2/-f .  (2 

In  fact,  the  integral  on  the  left  of  2)  is  uniformly  convergent  in  Sd  =(«,  jS),  since 

/).2  0.2 

^,<-^,        in  33. 

'\Ve  may  obviously  differentiate  1)  n  times,  which  gives 

rx^-e-y^^-dx  =  y^  .  ^  .  5  ...  2-^^^li  2/-^,        2/>0.  (3 

Jo  222  2       -^      -    '         •'^  ^ 


DIFFEKEJSTIATION  499 

690.    FresneVs  Integrals. 

Let  us  start  with  the  relation  689,  1), 

/.CO  /~ 

j    ey^-dx  —  ^^^y-i,        y>0.  (1 

Let 

/(^2')=^'  forx>0; 

=  0,  for  x  =  0. 

Then 

since  the  integral  on  the  right  is  convergent,  by  646.     We  can  invert  the  order  of 

integration  here,  by  680,  1.     For,  f(xy)  is  continuous  in  i?  =  (OcoOoo),  except  on 

the  line  x  =  0.     It  has,  moreover,   no   point   of   infinite  discontinuity  in  B.     The 

integral 

1    fdx=  \     — -^dx 
Jt)  ■  Jo    e'^'y 

is  uniformly  convergent  in  any  (0,  ^)  except  at  ?/  =  0.    The  integral 

is  uniformly  convergent  in  any  (0,  6),  except  at  x  =  0.    Finally, 

is  uniformly  convergent  in  33.     For 

r.         rx^sin^^^cos^y,  ^3 

Jo  L        {\  +  x*)e''y   Jo 

Hence 

^^p_gx__px^siny  +  cos?/^^^  -^ 

Ju    1  +  x*     Jo       (l  +  x*)e''* 

Here  I'l  is  uniformly  convergent  in  i8,  since  it  is  independent  of  y.    Likewise  T^ 
is  uniformly  convergent,  since  its  integrand  is  numerically 


< 


l  +  x2 
l  +  x* 


Thus  all  the  conditions  of  680,  1  being  fulfilled,  we  can  invert  the  order  of  inte- 
gration in  2),  which  gives  ^         ^ 

J^^^dx^Jdy 

=r^,  (4 

Jo    l  +  x* 

as  is  seen  from  .3),  on  passing  to  the  limit  y  =  oo.    But 

p        dX        ^  TT  1  TT 

Jo    1+a-t      4sin7r/4      2\/2 


500  INFINITE   INTERVAL   OF   INTEGRATION 

This  by  2),  4),  gives 

r^j^dy  =  ^.  (5 

>     V?/  \/2 

If  instead  of  multiplying  1)  by  sin  y,  we  had  multiplied  by  cosy,  we  would  have 
got  by  the  same  reasoning  _ 

r^J^dy  =  ^.  (6 

Jo  y/y  V2 

The  integrals  5),  6)  are  known  as  FresneVs  integrals.    They  occur  in  the  Theory 
of  Light. 

If  we  set  y  =  x^,  these  integrals  give 

1    sin  x'^dx  =  \    cos  x^dx  =  \\/-k /I. 

691.    1.  Let  us  show  that  Stoke's  Integral 

B  =  \  cos  (a;^  —  xy)3x  (1 

satisfies  the  relation  ^  ^     , 

^  +  12/^=0.  (2 

This  fact  will  enable  us  to  compute  S  by  means  of  an  infinite  series. 
We  have  in  the  first  place, 

^=  f  Ik  sin  (x3  -  £cy)dx  (3 

dy      Jo 

by  683,  2,  since  the  integral  3)  is  uniformly  convergent  in  any  33  =  (a,  /3). 
In  fact,  using  the  transformation  of  the  variable  employed  in  657 

u  —  a;(x2  —  ?/),  (4 

\  x  sm  (x^  —  xy)dx=  \    — — , 

Jb  Jc     Sx^  —  y 

where  b,  c  are  corresponding  values  in  4). 

But 

xsmu      smu         xu      _  ^(„)^(„,  y). 


3x2  —  2/         y        3x2  —  2/ 

We  can  now  apply  661,  1,  replacing  x  in  that  theorem  by  u.    Thus  there  exists 

a  Co  such  that 

\  C    xsmudul  ^  „  =  „ 

I  Jc     3  x2  —  2/  1 
But  then  the  relation  4)  shows  that  there  exists  a  B  such  that 

I  X  sin  (x^  —  xy)dx  <  e, 
Jft  I 

for  any  h  ^B,  and  for  any  y  in  ^.     Hence  the  integral  3)  is  uniformly  convergent. 


ELEMENTARY   PROPERTIES   OF   B(u,  v),  T(u)  601 

To  find  the  second  derivative  of  S^  we  cannot  apply  683  to  the  integral  3).     For 

j  x^  cos  (x^  —  xy)clx 

is  not  even  convergent,  as  we  saw  657. 

We  may,  however,  apply  684,  2.     In  fact, 

I  a;2  cos  (x^  —  xy)dx  =  \    ^^-  —  V  ^Qg  ^^.s  _  xy)dx  +  |  j  cos  (x^  —  xy)dx 

=  i  [sin  (x3  -  xy)-]l  +  r 

=  1  sin  (63  _  hy)  +  Y. 

But  .00 

i  cos  (x^  —  xy)dx 

is  uniformly  convergent,  as  we  saw  663,  7. 
On  the  other  hand, 

Psin  (63  _  by)dy  =cos(63-6y)-cos(6«-6«)^ 

Ja  6 

which  i  0  as  6  =  CO.    Thus  684,  2  gives 

—  r  X  sin  (x3  —  xy)dx  =  ^^  \  cos  (x^  —  xy)dx.  (5 

dy  JO  3   Jo 

From  1),  3),  5)  we  have  2). 

2.  Before  leaving  this  subject,  let  us  show  the  uniform  convergence  of  the  inte- 
gral 3),  by  another  method. 

From  the  identity  09  09  9 

3.-3x2-yy3x2-y       y^ 

3x33x3         9x3' 
we  have 

\  x  sin  (x^  —  xy)dx  =  \       ^   ~  ^  sin  (x^  —  xy)dx  +  ^  \       ^  ~  ^  sin  (x^  —  xy)dx 
Jb  Ji        3  X  3  Js        3  x3 

t  rsin(x3-x;/)^^^y       T        y 

Obviously  T3  is  uniformly  convergent  by  660,  2. 

That  Ti  is  uniformly  convergent,  was  shown  in  663,  6.    That  T2  is  uniformly 
convergent,  follows  from  661,  2  ;  since  Ti  is  uniformly  convergent. 

Elementary  Properties  of  B[u,  v),  T{u) 
692.    1.  In  641  we  saw 

B(w,  -y)  =  r    f'^^""  (1 

is  a  one-valued  function  whose  domain  of  definition  is  the  first 
quadrant  in  the  w-,  v-plane,  points  on  the  w-,  v-axes  excepted. 


502  INFINITE   INTERVAL   OF   INTEGRATION 

In  642  we  saw  ^^ 

r(M)=J    e-'^x'^-Hx  (2 

is  a  one-valued  function  whose  domain  of  definition  is  the  positive 
half  of  the  w-axis,  the  origin  excepted.  We  wish  to  deduce  here 
a  few  of  the  elementary  properties  of  these  functions. 

2.   By  a  change  of  variable,  the  integrals  1),  2)  take  on  various 
forms.     Thus  in  1)  set 

x=     - — 


1-^ 

We  get  />i 

B(w,  z;)  =  j  y-\\-yy-Hy.  (3 

If  we  set  here  -. 

y  =  \-z, 

we  get  /»i 

B(m,  v)  =  i   2^-1(1  -  zY-^dz.  (4 

In  3)  let  us  set  y  =  sin^  0 ;  we  get 

B(w,  v^  =  2  f'sin^"-!  0  cos^''-^  0  dO.  (5 

If  we  set  11/ 

X  =  log  \/y 

in  2),  we  get 

l\u)=  ]    log( 

3.  We  establish  now  a  few  relations  for  the  B  functions.      In 
the  first  place  the  comparison  of  3),  4)  gives 

B(w,  V)  =  B(v,  m),  (7 

which  shows  that  B  is  symmetric  in  both  its  arguments. 

As  addition  formulae  we  have  the  three  following  8),  9),  10), 

B(m  +  1,  vj  +  BCw,  z;  +  l)=  B(w,  v)  (8 

For,  pi 

B(w,  w)=  I   x'^-Hl  -  xy-\\  -  X  +  x)dx 

=  f  x^^^l-  xy-^dx  +  Cx^'Xl  -  xydx, 

which  is  8). 

i;B(w  +  l,  v)  =  mB(w,  ?j+ 1).  (9 


[  logn  dy.  .         (6 


ELEMENTARY   PROPERTIES   OF   B(m,  v),  T{u) 


503 


For, 


B(m  +  1,  v)  =  r  a:«(l  -  xy-'^dx; 


integrating  by  parts,  =  [  -  ^^ — ^7  +  -  C^'^'K^  -  xYdx 

L  V        Jo      v^o 


u 


which  is  9). 

From  8),  9)  we  have 


=  -B(w,  i;  +  l), 


B(w,  v)  =  — ' —  B(u,  V  -\-l)  =  — ^!—  B(w  +  1,  v}. 


(10 


We  can  show  now  that  B(w,  n)  =  B(?i,  w)  is  a  rational  function 

of  u,  viz.  :  Tj/       i\      1  /  i'11 

B(w,  1)  =  1/u.  (11 


B(m,  7l)  = 


11  2  w-1 


w     w  +  1     u  +  2         u-\-n  —  l 


(12 


For, 


B(w,  1)=  rx"-^dx  = 


which  proves  11).     From  this  we  get  12),  using  10). 
4.    We  establish  now  a  few  relations  for  the  V  function. 

For,  integrating  by  parts, 

x''e-'dx=\  -e-^x""       +u)    e-'x^'-'^dx 
0  |_  Jo  •^0 


(13 


Hx. 


We  observe  next  that 
For, 


=  1. 


r(i)  =  i. 

r(l)=  f  e-^dx=    -e- 

From  13),  14),  we  get 

T (u  -\-  n)=  u(u  -\-V)  •••  (u  +  n-  l)r(w); 

and  this  gives      „ ,  .     ^    „ 

^  r(w)=l-2-3  •••7i-l  =  n-l!, 

on  replacing  n  by  w  —  1  and  u  by  1. 


(14 


(15 
(16 


504  INFINITE   INTERVAL   OF   INTEGRATION 

A  formula  occasionally  useful  is 
1  1      r°7 

It  is  obtained  from  2)  by  replacing  a;,  by  ax. 

5.  The  r  function  is  continuous  for  any  w>0.     This  follows 
from  669,  1  and  663,  Ex.  1. 

The  derivative  is  given  by 

T'(u)=  1    e-'^x^'-^logxdx,         u>0.  (18 

fc/0 

This  follows  from  683,  2.     Similarly 

J ■•00 
e-''x''-nog^xdx,         w>0.  (19 

We  can  now  get  a  good  idea  of  the  graph  of  r(w).     In  fact,  the 
expression  2)  shows  that  r(M)  >  0  for  all  w  >  0. 
From 

r(«)=X  +X 

we  see  that  T>^^     -r^^  ^       . 

It  lim  1  (w)=  +00. 

«=0 

From  13)  we  see  that 

lim  r(M)  =  +  Qo. 

w=H-oo 

From  19)  we  see  that  r"(w)>0,  and  hence  the  graph  of  r(w) 
is  concave. 

Since  r(l)  =  r(2),  the  curve  has  a  minimum  between  1  and  2. 

^^'^^^^^'''  1.46163... 

6.  We  establish  now  the  important  relation  connecting  the  B 
and  r  functions,  t^^  ^t-,^  n 

B(«,.)  =  I$^.  (20 

From  17)  we  have 


(1  + 


i- —  =  :f7-^ re-''^y>^x''+''-'dx. 


ELEMENTARY   PROPERTIES   OF   B(u,  v),  T(u)  505 

Hence  by  1) 

B(w,  v)  =  f      y'\    ^y    =,—^ — .^  Cdy  rV+*'-y-^e-(i+^>^c?a:.     (21 

We  may  invert  the  order  of  integration,  by  680,  3. 
For,  in  the  first  place 

f{xy)=      ^(1+y)^ 

is  continuous  in  i2  =  (OooOoo),  except  on  the  lines  a;  =  0,  ?/  =  0. 
Secondly, 

is  uniformly  convergent  in  any  (a,  /3),  a  >  0,  by  663,  Ex.  1. 
Thirdly, 

is  uniformly  convergent  in  any  (a,  5),  a  >  0. 
Finally, 

exists.     For  in  ^_  ^^1% 

seta;?/=f,  a:>0.     Then 

Hence  for  a  >  0, 

(^rc  (    fdy  =  r(M)  I    e-^2;''-ic?a;. 

But  _ 

lim   1    e-^a;''-i(^a:  =  r(w). 

0=0    '^a 

Hence  X=  lira  X,  =  r(w)r(v).  (22 

a=0 

Thus  all  the  conditions  of  680,  3  being  fulfilled,  we  have  L=K. 
From  21),  22),  we  have  18). 


CHAPTER  XVI 
MULTIPLE   PROPER   INTEGRALS 

Notation 

693.  1.  In  Chapters  XII  and  XIII  the  theory  of  proper  inte- 
grals of  functions  of  one  variable  was  developed.  We  now  take 
up  the  corresponding  theory  with  reference  to  functions  of  several 
variables. 

2.  We  begin  by  explaining  a  notation  which  we  shall  system- 
atically employ  in  the  following,  and  which  is  similar  to  that  used 
in  the  earlier  chapters. 

Let  21  be  a  limited  point  aggregate  in  an  w-way  space  9?^.  Let 
f(xy,  '••  a;^),  or  as  we  shall  often  write  it, /(a;),  be  a  limited  func- 
tion defined  over  21.  Let  us  effect  a  rectangular  division  D  of 
space  of  norm  d.  To  simplify  matters,  we  shall  suppose  d  is  not 
taken  larger  than  some  arbitrarily  large  but  fixed  number.  Those 
cells  which  contain  points  of  21,  as  well  as  their  volumes,  will  be 
denoted  by  d^,  d^,  •••,  or  by  a  similar  notation.  Let  M,^  m^,  be  the 
maximum  and  minimum  of /(a;)  in  d^.     We  shall  set 

It  sometimes  happens  that  we  are  considering  points  of  two  or 
more  aggregates  21,  ^,  •••     Then  we  shall  write 

where  the  subscript  indicates  that  the  sums  1)  are  taken  over  the 
aggregates  2t,  ^,  •••  respectively. 

3.  We  shall  denote  the  maximum  and  minimum  of  /  in  21  by 

M  and  m  respectively.      The  greater  of  \M\  and  \m\  we  shall 

denote  by  F^  so  that 

l/(2^r-^JI<^.        in  21. 
506 


UPPER   AND   LOWER   INTEGRALS  507 

4.  The  oscillation  of /(a;^,  •••  a;,„)  in  the  cell  d^  is 

The  sum  _ 

is  the  oscillator^/  sum  of /for  the  division  D. 

Ujjper  and  Loiver  Integrals 

694.    The  sums  S^,  S^  form  a  limited  aggregate,  D  representing 
any  division  of  norm  <c?q;   moreover 


For 
Hence 
or 


m<m,<M^<M. 

tmd,  <  ^mji^  <  '2M^d^  <  l,Md, 

m^d^  <  S^  <Sj)<  Mld^,  (1 


"i* 


Since  21  is  limited,  the  cells  d^  are  all  contained  in  some  cube. 
Hence  1d^  is  less  than  some  fixed  number,  and  the  theorem  follows 
at  once  from  1). 

695.  1.  Let  f{x-^,  •••  x^^^O  in  21.  Let  I)  and  A  he  any  two  rec- 
tangular divisions  of  space.  Let  E  he  the  division  of  space  formed 
hy  superimposing  the  divisio7i  A  on  i>,  or  what  is  the  same,  the  divi- 
sion I)  on  A.      Then 

For,  let  d^  be  one  of  the  cells  of  B  which  is  subdivided,  on  super- 
imposing A. 

Let  ,       , 

denote  the  cells  of  LJ  in  d^  containing  points  of  2t.     Then,  to  the 
term  M^d^  in  S^,  corresponds  the  term 

in  Sp.     But 

"^""^^  ^Md<^Md<Md. 


508  MULTIPLE   PROPER   INTEGRALS 

2.   Similar  reasoning  shows  that: 

696.  1.  Letf{xy,  •••  x^)^  0  6e  limited  in  the  limited  aggregate  51. 
Let 

with  respect  to  all  rectangular  divisions  D.      Then 

lim  So  =  E.  (1 

(i=0 

Let  us  employ  the  graphical  representation  of  231.  The  points 
of  21  lie  in  a  certain  cube  S  of  edge  (7.  The  representation  of  S  is 
formed  of  m  segments  Sj,  •••  S^  on  the  Xy,  •••  x^  axes.  We  shall 
suppose  (5  taken  so  large  that  no  coordinate  of  any  point  of  31  is 
at  a  distance  <  2  (f^  from  the  ends  of  these  segments.  This  insures 
that  the  cells  d^  of  any  D  of  norm  <c?q  lie  within  g,  and  therefore 
that  Sc^^^C"". 

Since  S  is  the  minimum  of  all  S^^  there  exists  for  each  e  >  0  a 
division  A,  such  that 

;^<^^>^+e/2.  (2 

,  Let  D  be  an  arbitrary  division.     Let  us  superimpose  A  on  2), 
forming  a  division  E. 

The  division  E  is  formed  by  interpolating  certain  points,  let  us 
say  at  most  /i  points  in  each  of  the  segments  Sj,  •  •  •  (5^.  The  inter- 
polation of  one  of  these  points  may  be  interpreted  as  passing  a 
plane  parallel  to  one  of  the  sides  of  (S;.  Its  effect  is  to  subdivide 
certain  of  the  cells  of  (S.     The  volume  of  the  cells  so  affected  is 

Hence  the  superimposition  of  A  on  i),  being  equivalent  to  pass- 
ing at  most  w/i.  planes  parallel  to  the  sides  of  (5,  affects  cells  of  S 
belonging  to  the  original  division  i),  whose  volume 

V<miidC'^-\  (3 


UPPER   AND   LOWER   INTEGRALS  509 

Let  A  subdivide  c?,,  a  cell  of  D  containing  points  of  91,  into  the 
cells 

containing  points  of  21,  and  into  the  cells 
containing  no  point  of  51. 

I  IK 

where  R  denotes  the  sum  of  those  terms  common  to  Sq  and  S^, 
corresponding  to  cells  of  I)  unaffected  by  the  division  A. 

xience 

IK  LK 

Therefore 

0  <  ^^  -  ^^  =  ^{M,  -  M^^)d,^  +  ^MX. 

<mtidFC'^-\     by  3). 

If  we  take 

d'< 


2  nifiFC"'-^'' 

^^^^^®  Sj,<Sj,  +  e/2,         iorsinyd<d'.  (4 

But  regarding  F  as  formed  by  superimposing  D  on  A, 

Hence  2),  4),  5)  give 

or  _       _ 

which  proves  1). 

2.  A  similar  line  of  reasoning  shows : 

Let  f(x^  ■••  Xjn)<,  0  be  limited  in  the  limited  aggregate  91.     Let 

S  —  Max  Sji, 

with  respect  to  all  rectangular  divisions  I).      Then 

<i=0 


510  MULTIPLE   PROPER   INTEGRALS 

697.  Let  f(x-^  •■■  Xj^)  be  limited  in  the  limited  aggregate  %.     Then 

the  limits  i-      r^  to 

lim  Sj)^        iim  tSi) 

exist,  and  are  jinite. 

Let  us  take  c  >  0  so  large  that 

is  positive.     Let  M^,  iV^  be  respectively  the  maxima  of  /  and  g  in 
the  cell  d^.     Obviously, 

We  have  seen  in  696.  1  that 

lim  2iV^c?^,         lim  2cc?^ 
exist.     Hence 

lim  Sj)  =  lim  '2M^d^  =  lim  S(iV;  -  c)c?. 

=  lim  '^Nfil^  —  lim  %cd^ 
exists,  and  is  finite. 

To  show  that  i[^  ^ 

exists  and  is  finite,  we  introduce  the  auxiliary  function 

and  determine  c?>0  so  large  that  h  is  always  negative  in  51. 

698.  The  limits  S,  S,  whose  existence  was  established  in  697, 
are  called  the  lotver  and  upjjer  integrals  of  fi^x-^---  x,^  over  the 
field  21.      They  are  denoted  respectively  by 

J^/(«i  •••  ^m^d%  =J^/(2:^  ...  x„,^dx^  ••■dx^\ 

I_  1  (1 

J g/(a^i  •  •  •  ^™) ^51  =  Jgj/(«i  •••x^^dx^---  dx^. 

When  the  lower  and  upper  integrals  1)  are  equal,  we  denote 
their  common  value  by 

J^/(-^i  •■■Xm)d%=  J^i^i  ■  ■  ■  x,rddx^  -dx^',  (2 


Hence 

As 


UPPER   AND   LOWER   INTEGRALS  511 

it  is  called  the  integral  of  f  over  the  field  %.  In  this  case 
f(x-^  •••  a;,„)  is  said  to  he  integrahle  in  5t.  We  also  say  the  inte- 
gral 2)  exists. 

The  integrals  1),  2)  are  called  m-tuple  or  multiple  integrals. 

699.  1.  Let  f(x-^--- Xj„')  he  limited  and  integrahle  in  the  limited 
field  %.  Let  L  he  any  rectangular  division  of  norm  d,  and  f^  any 
point  of  2t  in  the  cell  d^.      Then 

Mm  y.fQ:)d  =  ffd%.  (1 

d=o  *>'2l 

Conversely,  if  this  limit  exists,  however  the  D's  and  ^'s  he  chosen, 
the  ujyper  arid  lower  integrals  of  f  are  equal,  and  f  is  integrahle. 

For 

r>i.<fit^<M, 

^mA<^Atyi.<^MA-  C2 

lim  "Imd^  =  I  , 

lim  t^.d,  =  r 

are  equal,  we  get  1)  on  passing  to  the  limit  c?  =  0  in  2). 

The  reader  will  observe  that  this  reasoning  is  precisely  similar 
to  the  first  half  of  the  demonstration  in  493.  The  second  half  of 
our  theorem  is  proved  by  a  reasoning  exactly  similar  to  the  second 
half  of  the  demonstration  of  493.  Instead  of  the  interval  h  —  a, 
we  have  here  a  cube  of  volume  0"\ 

2.  The  theorem  1  shows  us  that  we  may  take 

when  it  exists  as  a  second  definition  of  the  integral  of  f  over  31. 

700.  1.  The  theorems  of  495,  496,  497,  and  498  may  now  be 
extended  without  trouble  to  functions  of  several  variables.  For 
convenience  of  reference  we  restate  them  here  for  this  general 
case. 


512  MULTIPLE   PROPER   INTEGRALS 

2.  In  order  that  the  limited  function  fQc-^  •••  a:„,)  he  integrable  in 
the  limited  field  21,  it  is  necessary  and  sufficient  that  the  oscillatory 
sum  0.x)f  =  0,  as  the  yiorms  of  the  divisions  D  converge  to  0. 

3.  If  the  limited  function  f(^x^  ••■  a;,„)  is  integrable  over  the  limited 
field  21,  it  is  integrable  over  any  partial  field  of  21. 

4.  hi  order  that  the  limited  function  f(x-^  •••  a;^)  be  integrable  in 
the  limited  field  21,  it  is  necessary  and  sufficient  that  for  each  e>0, 
there  exists  a  division  D  for  which  the  oscillatory  sum 

5.  In  order  that  the  limited  function  f(x^  •••  2;,,^)  be  integrable  in 
the  limited  field  21,  it  is  necessary  and  sufficient  that,  for  each  pair 
of  positive  numbers  to,  a  there  exists  a  division  D,  such  that  the  sum 
of  the  cells  of  D  in  which  the  oscillation  off  is  >  oj,  is  <  <t. 

EXAMPLES 
■     701.   1.  Let  %  be  the  square  (0,  1,  0,  1). 

Let  f(x,  y)  =0,         for  x,  or  y  irrational; 

=  - ,       for  x  —  —  \  m,  n  relative  prime,  y  rational. 
n  n 

Then /is  integrable,  by  700,  5,  For, /is  >-  only  on  the  lines  «  =  !,  ^,  1,  |,  J, 
I,  I,  I,  I,  f,  •••,  the  denominators  of  the  fractions  being  ^q.    On  each  of  these 

lines  the  oscillation  in  any  little  interval  is  >-.     On  all  other  lines  the  oscillation 

1  ^ 

is  <-.     Obviously  there  exists  for  each  o-  a  division  for  which  the  sum  of  the 

squares  in  which  the  oscillation  is  >-  is  <(7  ;  and  the  integral  is  zero. 

2.  Let  %  embrace  the  points  x,  y  of  the  square  (0101)  for  which  x  is  rational. 
Let  '     /(x,  y)  =  -,        for  X  =  — ;  m,  n  relative  prime. 

Then  /  is  integrable  in  %,  as  the  above  example  shows. 


Content  of  Point  Aggregates 

702.  1.  We  extend  now  the  notion  of  content,  etc.,  considered 
in  514  seq.,  to  limited  aggregates  in  di^'  Let  us  effect  a  rectan- 
gular division  of  space  of  norm  B.     Let 

dp   0,2^   wg,    ••• 


CONTEXT  OF  POINT  AGGREGATES  518 

be  those  cells  containing  at  least  one  point  of  the  limited  aggre- 
gate 21 ;  while 

d[,    d'^,   d',    - 

denote  those  cells,  all  of  whose  points  lie  in  21. 

Then  the  limits 

M  =  lim  S<,         n  =  lim  -Zdl  (1 

6=0  £=0 

exist,  and  are  finite. 

For,  let  us  introduce  the  auxiliary  function  f(x^  ••  a^™)'  ^^'hose 
value  shall  be  0  in  9?„j,  except  at  the  points  of  2t,  where  its  value 
is  1.  Then,  using  the  notation  and  results  of  the  previous  articles, 
we  have : 

2t^  =  2itfX  =  2(^«, 


2l^=SmX  =  2< 

But  by  697, 

lim  ia,         lim  21^ 

5=0                          5=0 

ist,  and  are  finite. 

2.   The  numbers  21,  21  are  called  the  upper  and  lower  content  of  21 
We  have  thus : 


21 


=ffd%         2t=J/c?2l. 


When  2t  =  2t,  their  common  value  is  called  the  content  of  2t 

We  denote  it  by 

Cont  21, 

or  when  no  ambiguity  arises,  by  2t. 

To  be  more  explicit  it  is  often  convenient  to  set 

i  =  Coht  21,         i  =  Cont  21. 

A  limited  aggregate  having  content  is  measurable. 
Thus,  when  2t  is  measurable. 


Cont  2t  =  ffdU 


The  content  of  a  measurable  aggregate  in  ^^  is  called  its  area; 
ill  9^3  the  content  is  called  volume.  We  shall  also  use  the  term 
volume  in  this  sense,  when  w>  3.  : 


614  MULTIPLE   PROPER   INTEGRALS 

3.  As  immediate  consequence  of  the  reasoning  of  1,  we  have: 
Let  ^  be  a  partial  aggregate  of  21.      Then 

703.  By  the  aid  of  the  auxiliary  function  employed  in  702 
we  can  state  at  once  criteria  in  order  that  21  is  measurable. 

1.  For  21  to  he  measurable^  it  is  necessary  and  sufficient  that  the 
sum  of  the  cells  contairiing  both  points  of  21,  and  points  not  in  21  con- 
verge to  0,  as  the  norm  of  the  division  =  0. 

This  follows  from  700,  2. 

2.  In  order  that  21  be  measurable,  it  is  necessary  and  sufficient 
that  for  each  e  >  0,  there  exists  a  division  such  that  the  sum  of  the 
cells  embracing  both  points  of  21  and  not  of  %  is  <  e. 

This  follows  from  700,  4. 

Frontier  Points 

704.  1.    The  frontier  ^  of  any  aggregate  21  is  complete. 

For,  let  j?  be  a  limiting  point  of  ^. 

Then  in  any  I)^*(p'),  there  are  points  of  ^.  If  f  is  such  a  point, 
there  are  points  not  belonging  to  21  in  any  D^*{f).  We  may  take 
t]  so  small  that  B^  lies  in  D^.     Hence  jt?  is  a  frontier  point  of  21. 

2.    Let  2t  and  SQ  be  two  point  aggregates.     Let 


D  =  Dist  (x,  y')  =  V(2;i  -  y^^  +  •••  ^  {x^-  y^f 

be  the  distance  between  a  point  a;  of  2t  and  a  point  y  oi  ^.  Let  S 
be  the  minimum  of  J),  as  x  runs  over  21,  and  y  runs  over  ^.  Then 
8^0,  and  is  finite.     We  say  h  is  the  distance  of  21  from  ^,  and 

"^^'^^  S=Dist(2[,«). 

In  certain  cases,  21  may  reduce  to  a  single  point  a. 

3.    If  %,  Sd  are  limited  and  complete,  there  is  a  point  a  in  21,  and 
a  point  b  in  SS-,  such  that 

Dist  (a,  b)  =  Dist  (21,  «). 

If  Dist  (21,  ^)  >  0,  the  two  points  a,  b  are  frontier  points. 


DISCRETE   AGGREGATES  515 

For,  we  may  regard  x^---  a:^,  y^---  ym,  as  the  coordinates  of  a 
point  z  in  a  2  m-way  space  9?2to-  We  form  an  aggregate  (S  whose 
points  z  are  obtained  by  associating  with  each  x  of  31,  every  y  of 
SQ.  Then  the  domain  of  definition  of  Dist  (a;,  y)  in  2,  considered 
as  a  function  of  2w  variables,  is  precisely  S.  To  represent  (5  we 
may  employ  2  m  axes,  as  in  231.  Obviously  <^  is  limited  and  com- 
plete, since  21  and  ^  are. 

Then  by  269,  2,  there  exists  a  point  (a^  •••  a^,  6^  •-•  6^)  in  S,  at 
which  I)  takes  on  its  minimum  value.     Then 

are  the  points  whose  existence  was  to  be  proved. 

The  points  a,  h  ure  frontier  points  of  21  and  ^  respectively.  For, 
if  they  were  inner  points,  the  distance  between  D^(a)  and  D^ih^ 
equals  -^.^^  (a,  5)  -  2  8  <  Dist  (a,  5). 

4.  Let  ^  be  a  partial  aggregate  of  21.  If  the  distance  between 
the  frontiers  of  21  and  ®  is  not  0,  we  say  53  is  an  inner  partial 
aggregate  of  21 ;  also  21  is  an  outer  aggregate  of  Sb. 

Discrete  Aggregates 

705.    1.  Definition.     An  aggregate  of  content  0  is  discrete. 
Obviously,  if  Cont2l=0, 

21  is  discrete. 

2.  Every  limited  point  aggregate  of  the  first  species  is  discrete. 
Let  21  embrace  at  first,  only  a  finite  number  of  points,  say  n  points. 
Let  us  effect  a  cubical  division  of  space  of  norm 


m  I — 


such  that  the  points  of  21  lie  within  their  respective  cells.     Then 
the  sum  of  the  cells  containing  the  points  21  is 

^v^  <  nS"'  <  €. 

Thus  21  is  discrete,  and  the  theorem  is  true  for  aggregates  of 
order  0.  Let  us  therefore  assume  the  theorem  is  true  for  aggre- 
gates of  order  n  —  1  and  show  it  is  true  for  order  n. 


516  MULTIPLE   PROPER   INTEGRALS 

By  265,  21^"^  embraces  only  a  finite  number  of  points,  say 


^1'   ^2  '"  ^s' 

We  can,  as  just  seen,  inclose  these  within  cells  of  total  volume 
<e/2.  The  points  of  21  not"  in  these  cells  form  an  aggregate  ^ 
of  order  n—1.  By  hypothesis  we  can  effect  a  division  of  space, 
such  that  the  total  volume  of  the  cells  containing  both  points  of 
^  and  not  of  ^  is  <  e/2.  Thus  the  division  formed  by  superim- 
posing these  two  divisions,  is  such  that  the  volume  of  the  cells 
containing  both  points  of  21  and  not  of  21  is  <  e. 

706.  Let  '^  he  a  limited  aggregate  whose  frontier  points  "^  form  a 
discrete  aggregate.      Then  2t  is  measurable. 

For,  using  the  notation  of  702,  the  volume  of  those  cells  of  a 
division  i),  containing  both  points  of  21  and  not  of  21,  is 

where  ^^  is  the  volume  of  those  cells  containing  at  least  a  point 
of  '^.     But,  as  ^  is  discrete, 

Hence,  by  703,  1,  21  is  measurable. 

707.  1.  Let  9^?^  be  an  w-way  space.  Let  us  give  certain  of  the 
coordinates  of  a;  =  (2:p  •••  a;,„)  fixed  values.  For  example,  let 
^p+\  =  (^p+\^  •■■  ^m=(^m-  The  aggregate  of  points  x=(x^,  •■•  Xp,  a^+i, 
•••  a^)  may  be  regarded  as  constituting  a  p-way  space.,  9?^  ly^'f^g 
in  9?^.  The  point  x^  when  considered  as  belonging  to  9?^,  may  be 
denoted  more  shortly  by  x=^(x^.,  •••  Xp). 

2.  Let  %he  a  limited  aggregate  lying  in  9?^.  Lf  ive  consider  21  as 
lying  in  an  m-way  space  9?„i,  m  >p,  it  is  discrete. 

For,  let  21  lie  in  a  cube  (7,  of  volume  O,  in  9?^,  so  large  that  none 

of  the  points  of  21  come  indefinitely  near  the  sides  of  C.     Then  the 

upper  content  of  21,  relative  to  9?p,  is  <  C.     We  can  effect  a  division 

D  of  dim  of  norm  d  such  that  the  points  of  21  lie  within  the  cells  of 

L>.     Then  the  volume  of  all  the  cells  containing  points  of  21  is  less 

than  ^7^  „ 

Cd'^-p, 

which  converges  to  0,  with  d. 


DISCRETE   AGGREGATES  517 

708.  1.  Let  y=f(^x^,  ■••  x^  be  defined  over  an  aggregate  51. 
Let  x  =  (x^,  •■■  a;„j),  x'  =(x-^-\-h-^,  •••  x^^  +  h^^  be  two  points  of  21. 
The  increment  that  /  receives  when  x  passes  to  x'  we  have  denoted 
by  A/.     Let  us  set 


Ax  =  Dist  (x,  x'}  =  VV  +  ...  +  AJ, 
and  call  ^  . 

A/ 

Ax 

the  total  difference  quotient  of  /.  The  point  a^  may  or  may  not  be 
restricted  to  remain  near  x ;  if  so,  it  will  be  stated. 

2.  Let  the  limited  functions 

have  limited  total  difference  quotients  in  the  limited  discrete  aggre- 
gate %.      Then  Sdi  the  y-image  of  21,  is  also  discrete. 

For,  let  us  effect  a  cubical  division  of  the  rc-space  of  norm  d. 

Since  the  difference  quotients  are  limited  in  21,  there  exists  a 

fixed  Gr,  such  that 

\Af\<da,         1  =  1,2,  ...n, 

as  X  ranges  over  any  one  of  the  cells  d^  of  D.  Hence  each  coordi- 
nate y^  remains  in  an  interval  of  length  <  dGi  as  x  ranges  over  the 
points  of  21  ill  d^.  Therefore  y  =  {yi,  •••  ^„)  remains  within  a  cube 
of  volume  cZ"(t".     Hence  the  points  of  ^  have  an  upper  content 

<  Id'^a^'  =  dPa^'td'^  =  d^G-^'^O' 
S3  21 

Aslimi^  =  0,  Cont«  =  0. 

3.  As  a  corollary  of  2  we  have : 
In  the  region  R  let 

have  limited  first  partial  derivatives. 

Let  %he  a  limited  inner  discrete  aggregate.      Then  ^,  the  image 
of  21,  is  discrete. 


518  MULTIPLE   PROPER   INTEGRALS 

4.  Let  the  limited  functions 

I/l=fl(^V   •■■  O'   •••  ^n=fn(Xv   •••  ^m)  U  =  m -\- p  >  OU 

have  limited  total  difference  quotients  in  the  limited  aggregate  31, 
except  at  points  of  a  discrete  aggregate  A.  In  the  cells  of  any  cubical 
division  of  norm  d  <  c?^,  let  at  least  m  of  these  difference  quotients 
remain  limited.      Then  the  image  ^  of  %  is  discrete. 

For,  consider  one  of  the  cells  c?^,  containing  a  point  of  A.  At 
least  m  of  the  coordinates  of  a  point  y  remain  in  intervals  of  length 

<ad. 

All  we  can  say  of  the  other  coordinates  of  y  is  that  they  remain 
in  intervals  of  length  2F,  where  |/J<i^,  i=  1,  2,  •••  n.  Thus  the 
image  of  the  points  of  %  in  the  cells  d^  has  an  upper  content 

<  i(ady(2  Fy  =  a"'  2^  F^id^^  =  2^  Fpa"'E^  <  e/2, 

if  c?Q  is  taken  small  enough. 

The  content  of  the  image  of  the  other  cells  d^  is 

<td''a''<dPa''%o. 

K 

As  jt?  ^  1,  we  can  take  d^  sufficiently  small,  so  that  the  content  of 
these  cells  is  <  e/2. 

709.  An  important  class  of  discrete  aggregates  is  connected 
with  functions  having  limited  variation,  which  we  now  define. 
Cf.  509  seq. 

Let /(ajj  •••  x^')  be  limited  in  the  limited  aggregate  21.  Let  D 
be  a  cubical  division  of  space  of  norm  dKd^.  Let  the  oscillation 
of  /  in  the  cell  d^  be  (o^.     If  there  exists  a  number  w  such  that 

Sft)^cZ'«-i<w,  (1 

however  D  is  chosen,  we  say  that  f(x-^^  •■•  2;„,)  has  limited  variation 
in  %  ;  otherwise  it  has  unlimited  variation. 
From  1)  we  have 

Sa,,<--^.  (2 


PROPERTIES   OF   CONTENT  519 

710.  Let  the  limited  functions 

Vl  =/l(^l  •••  ^m)  •••  Vn-l  =fn-l(Xl  -"^m)  U  =  171  +  p  >  m 

have  limited  total  difference  quotients  in  the  limited   aggregate  21- 
Let  _ 

have  limited  variation  in  51.     As  x  =  {x-^  •••  x^  ranges  over  51,  let 
y  —  (^j  •••  ^„)  range  over  ^.      Then  ^  is  discrete. 

For,  let  us  effect  a  division  of  the  2;-space  of  norm  d.  Then 
Vv  '"  Vn-x  remain  in  intervals  of  length  <_dGr  as  x  ranges  over  the 
points  of  51  in  one  of  the  cells  d^.  Thus  if  to^  is  the  oscillation  of 
/„  in  c?^,  the  point  y  remains  in  a  cube  of  volume 

when  X  ranges  in  d^.     Thus  the  upper  content  of  ^  is 

Kd^a^-^io,     by  709,  2). 
As  this  converges  to  0  as  c?  =  0,  ^  is  discrete. 

Proijerties  of  Content 

711.  1.  Let  5t  be  a  limited  aggregate.  With  the  points  of  51 
let  us  form  the  partial  aggregates  %^,  •••  51^,  such  that  the  aggre- 
gate of  the  common  points,  or  of  the  common  frontier  points,  of 
any  two  of  these  aggregates  is  discrete.  We  shall  say  that  we 
have  divided  51  hito  the  unmixed  aggregates  5li,  •••  5tr  Also,  5t  is 
the  union  of  51^,   •••  51^. 

2.  Let  the  limited  aggregate  51  he  divided  in  the  unmixed  aggre- 
gates %^.  5I2'  •••  "^s-      Then 

i  =  5lj  +  ...+f,;    5l  =  ii  +  -+i,. 

For,  let  i)  be  a  rectangular  division  of  norm  8.  Let  ^^  be 
the  volume  of  all  those  cells  of  D  which  contain  points  of  more 
than  one  of  the  aggregates  51^,  •••  51^.  Let  51^,/)  be  the  volume  of 
those  cells  containing  points  of  51^,  t  =  1,  2,  •••  s.     Then 


620  MULTIPLE   PROPER  INTEGRALS 

Now,  by  hypothesis,  _ 

lim  ^o  =  0- 

6=0 

Hence  passing  to  the  limit  in  1),  we  get 

The  other  half  of  the  theorem  is  similarly  proved. 

3.  If  the  aggregate  51  can  he  divided  into  the  measurable  unmixed 
aggregates  ^l^,  •••  St^,  it  is  measurable,  and 

Cont  21  =  Cont  Ij  +  -  +  Cont  21,. 

This  follows  as  corollary  of  2. 

4.  Let  2lp  •••  %s^e  limited  aggregates  whose  frontiers  are  discrete. 
Let  21  be  the  union  of  these  aggregates.      Then  21  is  measurable^  and 

Cont  21  =  Cont  2li  +  •••  +  Cont  %,. 

For,  we  may  divide  2t  into  2ti,  •  •  •  21^,  and  these  latter  aggregates 
are  unmixed,  by  hypothesis.  The  aggregates  2li,  •••  21^  are  also 
measurable  by  706. 

712.  1.  Connected  with  any  limited  complete  aggregate  21  of 
upper  content  2t  >  0  is  an  aggregate  ^,  obtained  from  21  by  a  pro- 
cess of  sifting  as  follows  : 

Let  i)j,  i>2'  "■  b®  ^  S6t  of  rectangular  divisions  of  space,  each 
formed  from  the  preceding,  by  superimposing  a  rectangular  divi- 
sion on  it.     Let  the  norms  of  these  divisions  converge  to  0. 

The  division  B^  effects  a  division  of  21  into  unmixed  partial 
aggregates.  Let  %^  denote  those  partial  aggregates  whose  upper 
content  is  >  0.     Then,  by  711,  2,  %  =  %. 

Similarly,  the  division  D^  defines  a  partial  aggregate  of  21^  and 
hence  of  21,  such  that  %^  =  21,  etc.  Let  us  consider  the  cells  of  i)„ 
which  contain  points  of  2l„.  As  ti  =  oo,  these  cells  diminish  in 
size,  and  in  the  limit  define  a  set  of  points  ^.  The  upper  content 
of  the  points  of  21  in  the  domain  of  any  point  of  ^  is  >  0.  Thus 
each  point  of  ^  is  a  limiting  point  of  21,  and  hence  a  point  of  %. 
We  shall  prove,  moreover,  that  ^  is  perfect. 


I 


PROPERTIES   OF   CONTENT  521 

For,  suppose  h  were  an  isolated  point  of  ^.  Let  (7  be  a  cube 
whose  center  is  h  and  whose  volume  is  small  at  pleasure.  Let  a 
be  the  points  of  21  in  C.     Let  us  divide  Q  into  smaller  cubes,  say 

of  volume  -a.     The  points  of  31  in  at  least  n  of  these  new  cells 
n 

must  have  an  upper  content  >  0.     Thus  there  are  other  points  of 

^  in  O  besides  h.      Hence  ^  has  no  isolated  points.      To  show 

that  ^  is  complete,  let  ^  be  a  limiting  point  of  :^ ;  it  is  therefore 

a  point  of  31.     The  upper  content  of  the  points  of  %  in  any  domain 

of  /3  is  >  0.     /3  will  therefore  lie  in  one  of  the  cells  of  2>„,  w  =  1, 

2,  •••.     Hence  it  is  a  point  of  ^. 

Finally,  _     

For,  any  cell  of  i)„  which  contains  a  point  of  ^  contains  a  point 
of  2l„,  and  conversely  -dnj  cell  which  contains  a  point  of  2l„  con- 
tains a  point  of  ^,  or  is  at  least  adjacent  to  such  a  cell. 

2.  The  aggregate  ^  may  be  called  the  sifted  aggregate  of  %. 

713.  1.  We  shall  find  it  useful  to  extend  the  terms  cells,  division 
of  space  hito  cells,  etc.,  as  follows  : 

Let  us  suppose  the  points  of  any  aggregate  2t,  which  may  be 
9?^  itself,  arranged  in  partial  aggregates  which  we  shall  call  cells, 
and  which  have  the  following  properties : 

1°.  There  are  only  a  finite  number  of  cells  in  a  limited  portion 
of  space. 

2°.   The  frontier  of  each  cell  is  discrete. 

3°.  Each  cell  lies  in  a  cube  of  side  ^  S. 

4°.  Points  common  to  two  or  more  cells  must  lie  on  the  frontier 
of  these  cells. 

We  shall  call  this  a  division  of%of  norm  S. 

2.  Let  A  be  such  a  division  of  space.  Let  21  be  a  limited  aggre- 
gate. As  in  702,  %^  may  denote  the  content  of  all  the  cells  of  A 
which  contain  at  least  one  point  of  21 ;  while  2t^  may  denote  the 
content  of  those  cells  all  of  whose  points  lie  in  21. 


522  MULTIPLE   PROPER   INTEGRALS 

3.  Let  21  he  an  aggregate  formed  of  certain  of  these  cells.,  2lj,  •••  21^. 
Then  21  is  measurable  ;  and 

Cont  21  =  Cont  2li  +  •••  +  Cont  21,. 

This  is  a  corollary  of  711,  3. 

714.  Let  %he  a  limited  point  aggregate.,  and  A  a  division  of  space 
of  norm  S,  not  necessarily  a  rectangular  division.      Then 

limi^  =  I,         lim2l^  =  2l.  (1 

6=0  6=0 

Let  us  prove  the  first  half  of  1);  the  other  half  is  similarly- 
established. 

For  each  e  >  0  there  exists  a  cubical  division  D  of  norm  c?,  such 

*^^*     •  i<i^<i  +  e/2.  (2 

Let  D'  be  another  cubical  division  of  norm  d' . 
Let  ^ff  denote  the  volume  of  all  those  cubes  containing  points 
of  2li).     We  can  choose  d'  so  small  that 

ii,<^Z)'<  1^  +  6/2. 

Then  2)  erives  _     _        _ 

%<^D<%  +  e.  (3 

Let  A  be  any  division  of  space,  not  necessarily  cubical,  of  norm 
h<ld'._ 

Then  21^  contains  every  point  of  21 ;  and  is  a  part  of  ^^',  since 
the  distance  of  21  to  ^^^  is  ^  c?'.     Hence,  by  702,  3,  and  713,  3, 

i<iA<«z,'. 

This  srives  with  3)  _     _       _ 

2l<2l^<2l4-e 

for  any  h<\d' . 

715.  1.  Let  21  be  a  limited  aggregate.  If  21  is  not  complete.,  let 
us  add  to  it  its  lacking  limiting  points.  The  resulting  aggregate 
^  nia}^  be  called  the  completed  aggregate  of  21. 

A  limited  aggregate  21,  and  its  completed  aggregate  ^,  have  the 
same  upper  content. 


PROPERTIES  OF  CONTENT  523 

For,  let  us  effect  a  rectangular  division  D  of  norm  d.  The  cells 
containing  points  of  ^  fall  in  two  classes :  1°,  those  cells  c^j,  d^,  ••• 
containing  points  of  2t ;  2°,  those  cells  e^  e^^  •••  containing  no  point 
of  21-  Each  of  these  latter  cells,  as  e^,  is  contiguous  to  at  least  one 
cell  d^.  If  gi,  •••  are  contiguous  to  c?^,  we  will  join  them  to  c?^,  to 
form  a  new  cell  S^,  in  such  a  way  that  each  e-cell  has  been  joined 
to  some  one  c?-celL 

The  cells  Sj,  h^^  •••  together  with  the  cells  c?j,  d^^  •••  which  remain 
unchanged  by  this  process  of  consolidation,  define  a  division  A 
of  the  kind  considered  in  713.  The  norm  h  of  this  division  is 
evanescent  with  d. 

Now,  for  the  division  A, 

By  714,  the  left  side  =  %.     Hence 

«  =  i. 

2.  The  lower  contents  %,  ^  do  not  need  to  be  equal. 

For  example,  let  21  =  rational  points  in  the  interval  J"=  (0,  1). 
Then  ^  =  J". 

^"^  21  =  0,         «  =  1. 

3.  Let  21  be  measurable.  Then  21,  and  its  completed  aggregate  S, 
have  the  same  content. 

For,  we  have  just  seen  that 

21  =  21  =  ^.  (1 

On  the  other  hand,  every  inner  point  of  21  is  an  inner  point 
of  ^.     Hence 

Hence,  passing  to  the  limit, 

2l  =  2t<«<^.  (2 

Hence  1),  2)  give  ^^^^^^ 


524  MULTIPLE   PROPER  INTEGRALS 

4.  If  %  is  measurable^  the  content  of  21  and  its  derivative  %'  are 
equal. 

For,  let  ^  be  the  completed  aggregate  of  31.  Since  every  inner 
point  of  21  is  an  inner  point  of  21',  and  every  point  of  21'  is  in  ^, 
we  have  for  any  cubical  division  7),  of  norm  d^ 

Passing  to  the  limit  c?  =  0,  this  gives,  since  21  is  measurable, 

21<2['<21'<^.  (3 

But,  by  3,  21  =  ^.     Hence  3)  gives 

21  =  f['  =  W. 

716.  Let  %  be  a  limited  aggregate  whose  upper  content  is  21.  Let 
^  be  a  partial  aggregate  depending  on  u  such  that 

lim  ^  =  2t. 

Let  D  be  a  rectangular  division  of  norm  d.  Then  for  each  e  >  0 
there  exists  a  pair  of  numbers  Uq,  d^,  such  that 

%-'^u,D<e  (1 

for  any  0 <u<Uq,  0 < c? < c?g. 
For,  ii  d<  d^, 


and,  if  w  <  Mq, 


2l<2l^<2l  +  e/2; 
1  -  6/2  <  ^„. 


Thus  i_e/2<B„<^,,^<S^<i  +  6/2, 

which  establishes  1). 


Plane  and  Rectilinear  Sections  of  an  Aggregate 

717.  1.  Let  21  be  an  aggregate  in  $R„j.  As  x  =  {x^  ••■  a;,„)  ranges 
over  21,  x^  will  range  over  an  aggregate  j^  on  the  aj^-axis,  which  we 
call  the  projection  of%on  this  axis. 


PLANE  AND  RECTILINEAR  SECTIONS  OF  AN  AGGREGATE     525 

The  points  of  9?„j  for  which  one  of  the  coordinates  as  x^  has  a 
fixed  value  x^  =  ^^,  lie  in  an  m  —  1  way  plane,  which  we  shall  say 
is  perpendicular  to  the  x^-axis.  We  may  denote  it  by  P^  or  more 
shortly,  by  P,.  The  points  of  21  in  P^  form  a  plane  section  of  21 
corresponding  to  the  point  f^  in  j:^,  which  we  denote  by  ^^  or  '^^. 
We  also  say  ^^  is  a  plane  section  of  %  perpendicular  to  the  x^-axis. 

2.  As  x=  {x-^^---x^^  ranges  over  the  points  of  31,  the  point 
(a;j,  •  •  •  x^_i,  0,  x^+i,  •  •  •  a:^)  ranges  over  an  aggregate  2i,  in  the  plane 
x^  =  0,  which  ma}'-  be  called  the  w  — 1  tva^  plane  U^  of  the  axes  per- 
pendicular to  x^.     We  call  36„  the  projection  of%on  11,. 

3.  Let  us  fix  all  the  coordinates  of  a;  =  (a;^  •••  rc^),  except  x^. 
Then  x  describes  a  right  line  parallel  to  the  x^-axis.  Let  o,  denote 
the  points  of  21  on  one  of  these  lines.  We  shall  call  it  a  rectilinear 
section  of  21,  parallel  to  the  x-axis. 

4.  Let  21  be  limited  and  complete.  Then  the  ^^  and  the  a,,  also 
the  ]C,  and  ^,,  are  complete. 

Let  us  show  that  the  S^,  are  complete.     Let  j9  be  a  limiting  point 

in  one  of  the  'C,.     Let 

Pv  P2  •••  (1 

be  a  sequence  of  points  in  this  plane  which  =p.  Then  1)  is  a 
sequence  in  21,  and  as  21  is  complete,  p  lies  in  21,  and  hence  in  '^^. 

Let  us  show  that  j,  is  complete.  In  fact,  let  q  be  one  of  its 
limiting  points.  Let  q^,  q^  •■•  be  a  sequence  in  y,  which  =  q.  In 
each  plane  section  ^^  ,  take  a  point  r^. 

This  gives  a  sequence 

whose  limiting  points  lie  in  21,  since  2t  is  complete.  Moreover,  the 
projection  of  these  limiting  points  is  q. 

718.    1.  Let  %  he  a  measurable  aggregate. 

Let  jCg.  denote  those  points  of  j,,  for  which  the  upper  content  of  the 
frontier  points  of  ^,  is  ^  cr.      Then  j:^  is  discrete. 

For,  let  us  effect  a  cubical  division  _Z>  of  di^  of  norm  d.  This 
effects  also  a  division  of  norm  d  of  the  a;,-axis.  Let  d^,  d^  ••■  denote 
those  intervals  on  this  axis,  embracing  at  least  one  point  for  which 
the  frontier  points  of  the  corresponding  plane  section  have  upper 


626  MULTIPLE   PROPER   INTEGRALS 

content  >a-.     If  ^^  denote  the  volume  of  those  cells  containing 
frontier  points  ^  of  51,  we  have 

^2)  >  o'^d,,         for  any  D. 

Let  c?  =  0.     As  SI  is  measurable, 

a  Cont  ^^  =  0. 

As  o-  >  0,  Cont  j^  =  0. 

2.  In  a  similar  manner  we  prove  : 

Let  7i^  denote  those  points  of  the  projection  of  the  measurable  aggre- 
gate %  on  the  plane  x^  =  ^^for  ivhich  the  content  of  the  frontier  points 
on  the  corresponding  rectilinear  sections  is  ^  cr.      Then  Ti^  is  discrete. 

3.  Let  TC^  be  the  projection  of  the  measurable  aggregate  %  07i  the 
x^-axis.  Let  D  be  a  division  ofdl,n  of  norm  d.  Letf^^  fi'"  denote 
those  intervals  on  the  x-axis  contahmig  frontier  points  of  ^^.  Let 
7>0,  o->0  be  taken  small  at  pleasure.  If  f'li  f'i  ■••  denote  those 
f -intervals  contaiyiing  points  of  ^*^,  for  ivhich  the  upper  content  of  the 
corresponding  plane  sections  "^  is  ^  7,  we  can  take  d^  so  small  that 

^f[<(r,         dKd^. 

For,  in  the  contrary  case,  the  upper  content  ^  of  the  frontier 
points  of  21  is  =^^_  ^^ 

But  21  being  measurable,  0^  =  0,  which  contradicts  1). 

Classes  of  Integrdble  Functions 

719.  1.  Let  f(x-^  •■■  x„i)  be  conti7iuous  at  the  limiting  points  of  the 
limited  complete  field  21.      Then  f  is  integrable  in  21. 

For,  reasoning  similar  to  that  of  352  shows  that  we  can  effect  a 
cubical  division  2>,  such  that  the  oscillation  of  /  in  each  cell  of  D 
containing  points  of  21  is  <co.     Then  by  700,  4, /is  integrable. 

2.  In  the  limited  complete  aggregate  2t,  let  the  limited  function 
f{x^  •  •  •  Xjn)  be  continuous,  except  at  the  points  of  a  discrete  aggregate 
^.      Then  f  is  integrable  in  21. 

Since  ^  is  discrete,  there  exists  a  cubical  division  D  such  that 
the  volume  of  those  cells  containing  points  of  ^  is  <  e. 


CLASSES   OF   INTEGRABLE   FUNCTIONS  627 

Let  (5/)  denote  those  cells  which  contain  points  of  21,  but  do  not 
contain  points  of  ^.  Since  /  is  continuous  in  (5^,  we  can  effect  a 
cubical  division  jy  of  g^,  such  that  the  oscillation  of  /  in  any  cell 
of  i)'  is  <  ft). 

Then  by  700,  4,  /  is  integrable  in  %. 

3.  Let  f(^x-^  •  ■  •  x^  have  limited  variation  in  the  limited  field  %. 
Then  f  is  integrable  in  %. 

*'°^'  n^/=2ft>,(^.<^-5;ft,,, 

<da),        by  709,  2). 
H«^««  limn^/=0, 

and /is  integrable,  by  700,  2. 

720.  As  in  504,  505,  507,  and  508,  we  may  establish  the  follow- 
ing theorems  : 

1.  Letf(x-^  •••^rn)  ^^  ^  limited  integrable  function  in  the  limited 
field  21.      Then  \fC^i  •••  x„^)\  is  integrable  in  21.      [507.] 

2.  Let  fy,  fiy  ■■■  fr  ^^  limited  integrable  functions  in  the  limited 
field  21.     If  C\^  Ci",  •■■  Cf  denote  constants^  then 

are  integrable  in  21.      [504,  505.] 

3.  The  converse  of  1  is  not  necessarily  true.     For  example,  in 

a  rectangle  i2  let  /.^     x      ^  r  ,  •       i 

j{xy)  =  1,  tor  2:,  y  rational ; 

=  —  1,         for  other  points  in  R, 

Obviously /is  not  integrable  in  R. 

On  the  other  hand,  |/|  obviously  is  integrable. 

4.  The  product  /  •  g  may  be  integrable  without  either  /  or  g 
being  integrable  in  21.  For  example,  in  a  rectangle  R  let  f{xy) 
be  defined  as  in  3 ;  while 

g(xy^  =  —  1,         for  a;,  y  rational ; 

=  1,  for  other  points  of  R. 

Then^  =  —  1  in  ^,  and  is  hence  integrable  in  R. 


528  MULTIPLE   PROPER   INTEGRALS 

721.  Let  f(x-^  ••■  Xjn)  be  mtegrahle  in  the  limited  complete  field 
%.     Let  (^  he  the  points  of  %  at  which  f  is  continuous.     Then  g  =  51. 

For,  if  %  is  discrete,  the  theorem  is  true,  even  if  /  has  no  points 
of  continuity  in  51.  Let  us  therefore  suppose  31  >  0.  Let  ^  be 
the  partial  aggregate  formed  from  21  by  the  process  of  sifting, 
considered  in  712. 

Let  _Z>  be  a  rectangular  division,  and  d  one  of  its  cells  containing 
points  of  ^ ;  we  can  choose  _Z>  so  that  no  cell  has  points  of  ^  only 
on  its  sides.  Let  o  be  the  points  of  21  in  d.  Since  a  is  a  partial 
aggregate  of  21, /(a;^  •••  a;^)  is  integrable  in  o.  The  reasoning  of 
508  shows  now  that  /  must  be  continuous  at  one  point,  at  least, 
of  a  and  hence  at  an  infinity  of  points  of  a. 

Among  these  points,  lie  points  of  iB.  Thus  every  cell  of  the 
division  2>,  which  contains  a  point  of  ^,  contains  a  point  of  ^. 

Hence  I  =  f . 


Generalized  Definition  of  Multiple  Integrals 

722.  Letf(x^  •■■  x^')  be  liynited  in  the  limited  field  %.  Let  A  be 
any  division  of  space  of  norm  S  into  cells  8^  8^.,  •••,  not  necessarily 
rectangular.  Let  Wl^,  VX^  be  respectively  the  maximum  and  Tninimum 
off  in  S^.      Then  __ 

lim  E^  =  lim  ^m8^  =  Cfd%  (1 

6=0  5=0  »/2l 

lim  S^  =  lim  2m^a^  =  ffd^.  (2 

S=o  &=o  ^21 

Let  D  he  a,  cubical  division  of  norm  d.  Let  c?^,  d^.,  •••  be  the 
cells  of  D  containing  points  of  21.  We  may  denote  their  volum.es 
by  the  same  letters.  Let  M^  =  Maxf  in  d^;  also  #^Maxj/|  in 
21,  and  ^  1.     Then  for  each  e>  0,  there  exists  a  d  such  that 


^%\     2 


where,  as  usual,  ^       ^^.^  ^ 


GENERALIZED   DEFINITION   OF  MULTIPLE   INTEGRALS    529 
Furthermore  we  may  choose  d  so  small  that 

Sz)-I<g^,  (4 

where  21^  =  Xd^. 

Consider  now  the  division  A.  Those  of  its  cells  containing 
points  of  %  fall  into  two  classes :  1°,  those  lying  in  only  one  cell 
of  D;  2°,  those  lying  in  two  or  more  cells  of  D.  Let  8^^,  S^^,  •••be 
the  cells  of  the  1°  class  lying  in  d^.  Let  SJ,  ^2,  •••  be  all  the  cells 
of  the  2°  class.  Then  the  content  of  all  the  cells  of  A  containing 
points  of  %  is 

But  since  the  frontier  of  %q  is  discrete,  there  exists  a  8^  such 
that 

As  moreover  21^  =  2(,  by  714,  we  may  suppose  that 

From  5),  6)  we  have 

|2^..-i|<3^- 
This  with  4)  gives  finally 

Mow 

where  3)?^^,  3)Z[  are  the  maxima  of/  in  8^^,  8[  respectively.     Hence 

S^^tM^K  +  F^h[,  (8 

Hince  3U^..^^,         m[<F. 

Thus  5),  8)  give 

S^<tM^K  +  l-  (9 


530  MULTIPLE    PKOPLR    INTEGRALS 

Now 

<|,         by  7).  (10 

Thus  9),  10)  give 

^^<S^  +  l-  (11 

In  the  same  way  we  may  show  that  for  a  properly  chosen  cubi- 
cal division  F, 

*^^<^.  +  |-  (12 


From  3),  11),  12)  we  have 


S\ 


-f\<€,        B<B,. 


This  proves  1).     In  a  similar  manner  we  may  demonstrate  2). 

723.  LetfQx-^  •••  x„^')  he  limited  in  the  measurahle  field  21.  Let 
A  he  an  unmixed  division  of  2t  of  norm  S,  into  the  cells  Sj,  h^i  ••• 
As  usual  let  _ 

Let  m  he  the  maximum  of  S^^  and  3)^  the  minimum  of  S^  for  all 
divisions  A,  S  =  0.      Then 


Let  us  divide  one  of  the  cells  as  S^  into  two  unmixed  cells  8[, 
h'J .  This  gives  a  new  division  A'.  Then  the  term  m^h^  in  S^^  is 
replaced  by  the  two  terms 

m[h[  +  m['h['  ^  WjS^, 
in  S^'.     Hence 

Similarly 


PROPERTIES   OF   INTEGRALS  ^        531 

The  theorem  follows  now  from  722.    For,  there  exists  a  division 

A,  such  that  ^    ^ 

m  — e<A^^<m.  (1 

Let  us  now  take  a  sequence  of  divisions  A',  A",  •••  whose  norms 
=  0 ;  each  A^"^  being  formed  by  subdividing  the  cells  of  A^"~^^ 
Then  ^ 

S^^S^^S^ =  )fd%.  (2 

From  1),  2),  we  have   ,  , 

l/c^I-m  <€; 


hence 


m  =  I  jd%,        etc. 


Properties  of  Integrals 

724.  Let  f(x^  •••  Xj^)  he  limited  and  integralle  in  the  limited  field 
21.  Let  ^  he  a  partial  aggregate  depending  on  u,  such  that  ^  =  21, 
as  w  =  0.     Then 

lim    I  fd^  =  1  fd%.  (1 

Since  /  is  integrable  over  21, 

nj,f<€/2  (2 

for  any  division  D  of  norm  d  <  d^^  by  700,  2. 

Moreover,  by  716,  if  d^  and  Uq  are  taken  small  enough, 

2ti)-^«,i,<e/2^,  (3 

where  |/|<^in  21. 

Let  £?j,  c?2'  ■"  ^6  ^^^6  cells  of  I>  containing  points  of  :S9„,  and 
d'l,  d'2,  •••  the  cells  containing  onl}^  points  of  21.     Then 

S^^=^M^d,  +  XM[d[, 

S^^^^=2N^d,, 

where  iV^  =  Max  /  for  points  of  ^  in  d^. 

Hence  

\^%,-S^^,J<^CM^-nd.  +  F2d[ 

<e,     by  2),  3).  (4 


532  MULTIPLE   PROPER  INTEGRALS 

Let  now  d=0.     Then  4)  gives 

If-f  l<- 


Hence 


\S-S 


which  gives  1). 

725.  1.  Letf(x^  •••  x„^  he  limited  in  the  measurable  field  %.  Let 
^  he  an  outer  field  of  51.  Let  g{x^  ■■•  2;,„)  =  0  in  ^,  except  at  the 
points  of  21,  where  it  =zf(x-^  ■•■  x^').      Then 

J/dn=J^gcm;  £fm=£ffdss. 

For,  let  i)  be  a  division  of  space  of  norm  d,  not  necessarily 
rectangular.  Let  the  inner  cells  of  51  be  d^,  d^,  •••  while  d'l,  d'2,  •■• 
denote  cells  containing  frontier  points  of  51.  Let  M^^  N^  denote 
the  maxima  of/,  g  in  d^,  while  M[,  N[   are  the  maxima  of/,  g 

in  d'.     Then  -.       ^ 

Sj,  =  tMA  +  ^M[d[, 

Tj,  =  ^N^d^  +  tN[d[. 
Hence,  since  M^  =  N^,  we  have,  setting  |  /|  <  .F  in  51, 

\Sj,-T^\<F^d[. 

Let  now  d=Q.  Since  51  is  measurable,  we  have  the  first  part 
of  our  theorem.     The  second  part  follows  likewise. 

2.  In  a  similar  manner  we  establish  the  following  theorems : 

Let  f{x^  •■•  x^')  he  limited  in  the  measurable  field  51.  Let 
g(x-^  •••  Xjn)  =  f  at  inner  points  of  51,  and  =  0  at  frontier  points. 
Then  r  r  r  7^ 

3.  Let  /(a^i  ••■  a;^),  gC^i  ■■•  ^m)  ^^  limited  in  the  measurable  field 
5t.     Let  them  he  equal  except  at  the  points  of  a  discrete  aggregate. 


PROPERTIES   OF   INTEGRALS  533 

726.  1.  Let  f(x-^  ■••  x^)  be  limited  and  integrable  in  the  limited 
field  %.     Let  ^  be  a  partial  aggregate  of  21,  such  that  21  =  ^.     Then 

Let  i)  be  a  cubical  division  of  space  of  norm  h.  Let  d[  denote 
those  cells  containing  points  of  ^,  and  d'l  denote  the  cells  con- 
taining points  of  21,  but  not  of  ^.  Then  employing  the  usual 
notation, 

Viiyd=tfQ[yi[  +  ^M'nd':. 

Since  %d"  =  0  as  S  =  0,  we  have  1)  on  passing  to  the  limit. 

2.  Let  f{x-^  ■■■  Xj^  be  limited  and  integrable  in  the  limited  complete 
field  21.  Let  S  denote  the  points  of  %  at  which  f  is  continuous. 
Then 


r/«i=X/<is. 


This  follows  from  1  and  721. 

727.    Let  f{x^  •••  x^')  be  limited  in  the  limited  field  21.     Lf'  for  any 
cubical  divisions  2),  of  norm  dKd^^ 

then  r  'r 

A<jjm<j^fd%<B.  (1 

For,  in  each  cell  d^  there  are  points  2;[,  2;'/,  such  that 

/(o^lXm^  +  o-,    f{x'l')>M-a, 

however  small  cr  >  0  is  chosen. 
Hence 

A<tf(ix[~)d^<Sj,  +  cjtdr,    B^tf{a^!)d,>Sj,-atd..       (2 

Let  g 


CMt2l 


534  MULTIPLE   PROPER   INTEGRALS 

Then  passing  to  the  limit  d  —  0  in  2),  we  get 

A-e<  i  <  I   <B  +  e. 
From  this  we  conclude  1}  at  once. 

728.  Letf(x^---Xj^  he  limited  in  the  limited  field  %.  Let  51  he 
divided  into  the  unmixed  fields  51  j,  •••  51^.      Then 

ffM=ffd^i  +  -  +  ffd%s, 
ffd^=ffd%,  +  -  +  ffd%. 

For,  let  j/|<^in  31.  Let  D  he  a  rectangular  division  of  norm 
d.  As  in  711,  2,  let  5li,i),  •••  51^,^  be  the  cells  containing  points  of 
Slj,  •••  5ls,  respectively;  while  ^^  constitute  the  cells  containing 
points  of  more  than  one  of  the  fields  5li  •••     Then 

\^d-(:^i,d+--^^s,d)\<sF^^. 

Letting  c?  =  0  in  this  relation,  we  get  the  first  part  of  the 
theorem.     The  rest  is  proved  likewise. 

729.  As  in  504 ;  489,  4 ;  526,  2 ;  531,  we  may  prove  the  follow- 
ing theorems. 

1.  Letf^,  ••■fg  he  integrahle  in  the  limited  field  %.  Let  Cj,  •••  Cs  he 
constants  and  ^  „ 

Then 

fFdn  =  c.  f/M  +  •••  +  c,  ffM-  [504] 

2.  LetfQc-^  •••  Xj^  he  integrahle  in  the  limited  field  21,  and  numer- 
ically <M.      Then 

f  fdn  <  MCont  51.  [489,  4] 

3.  Letf{x^  ■••  a:^),  g(x-^  •••  x„,^  he  integrahle  in  the  limited  field 
51,  and  letf<g.     Then       ^  ^ 

\  fd%<\  gd^^.  [526.^1 


PROPERTIES   OF   INTEGRALS  585 

4.  Letf{x-^  •••  a;,„),  g(x-^  •••  x^^)  he  integrable  in  the  limited  field 

21.     Let 

/^O;         ©=Mean^,         in  21. 

Then 

(fgd%=^(fd%.  [531] 

5.  The  following  theorems  are  readily  proved : 

Letf(x-^   '•  x^~)  be  limited  in  the  limited  field  3l-    If  f(x-^  •••  a;^)>X 
in  21,  _ 

r/c^2l^Xl. 
^% 

6.  Let  f{x-^  •••  x^  he  limited   in   the   limited  field  21,  and  ^0. 
If  ^  is  a  partial  field  of  21, 

JfM^Jfd^. 

For,  >^2t^^A^53^. 

7.  Letfy,  ••■fn  he  limited  in  the  limited  field  21.      Then 

jSfl  +  -  +fn)d%  <(  f^d%+..-  +£fM. 

iiSl  ^l2l  il2l 

For,  in  any  cell  d^  of  the  division  D, 

Max(/i+   ..+/„)<Max/i+...  +  Max/„ 
Min(/i  +  -  +/„)^Min/i+  •..  +  Min/„. 

8.  Letf  g  he  limited  in  the  limited  field  2t.      Then 

J(f-g)d^^ffd^-fgd% 

*^2I  -^n  ^% 

f(f-g)d%^ffm-fgdn. 


536  MULTIPLE   PROPER   INTEGRALS 

For,  in  any  cell  d^  of  the  division  i), 

Max(/- g)  ^Max/-  Max g, 
Min(/-^)<  Min/-  Min  g. 

9.  Let  /,  g  he  limited  in  the  limited  field  31,  aiid  let  f<g.      Then 
ffd^<  fgd%;     Tfd%<  ffd'^. 

r^%  ^21  c/gt  ^gj 

730.  Let  f(x^  •••  Xjn)  be  limited  in  the   limited  field  31.     Let%^ 
denote  the  points  of  %  at  which  |/|>o-.     If  %^  is  discrete  for  any 

ifd'^=  ifdu  =  o. 

For,  let  _  ,      _        . 

(7<e/2l,         |/|<i^. 

Let  i)  be  a  division  of  norm  S.     Let  d'^  denote  the  cells  in  which 
|/|>o-,  while  d^l  denotes  the  cells  in  which  |/|<  cr.     Then 

^^°^^  \Js\<Fld'^-{-a^d'l. 

Let  S  =  0.     The  right  side  is 

<  o-  Coht  51  <  e, 
which  establishes  the  theorem,  by  727. 

731.  Letf(x-^  •••  x,n)  ^0,  he  limited  and  integrdhle  in  the  limited 

field  %.     If  V. 

/^3l  =  0, 

f  Ag  points  51(^1  <^^  ivhich  f(x)  >  a,  an  arhitrarily  small  positive  number^ 
form  a  discrete  aggregate.  Let  3  denote  the  points  at  which  f=  0. 
If  21  is  complete^  __     _ 

3  =  51. 

For, 

0=  r>  f,         by  729,  6. 


MULTIPLE   INTEGRALS   TO   ITERATED   INTEGRALS      537 
But 


r 


,/^^3I.,        by  729,  5. 

Ifence  __ 

^21,  =  0. 

Since  o-  >  0, 

%^  must  =  0. 

To  prove  the  second  part  of  the  theorem,  let  5  be  a  point  of  ^, 
the  sifted  aggregate  of  21,  at  which/ is  continuous.      Cf.  712. 

Then  if  />  0,  we  can  choose  3>0  so  small  that  />A,>0  in 
V^(h').  But  the  upper  content  of  the  points  a  of  2t  in  Fg  is  a>0. 
Hence 

Hence /=  0  at  every  point  of  continuity  of  21  in  ®.  Let  now 
7)  be  a  rectangular  division  of  space.  The  reasoning  of  721  shows 
that  every  cell  which  contains  a  point  of  ^  also  contains  a  point 
of  continuity  lying  in  ^.     Hence, 

which  gives, 


3^a     or       3  =  ^. 


Reduction  of  Midtiple  Integrals  to  Iterated  Integrals 

732.  1.  Let  f(xi  ■■■  x,„^  be  limited  in  the  limited  field  21.  Let 
j^  be  the  projection  of  2t  on  the  a;^-axis.  Let  ^^  be  a  plane  section 
of  21  perpendicular  to  the  a;^-axis.  Then  the  (m  —  l)tuple  upper 
and  lower  integrals  _ 

ffd^^,         ffd'^,  (1 

are  one-valued  limited  functions  of  x^,  defined  over  J^.  For,  let  21 
lie  in  a  cube  of  side  C.  Let  If^x^^  •••  a;^)|<^.  Then  both  inte- 
grals are  numerically 

for  any  x,  in  f ^. 


638  MULTIPLE   PROPER   INTEGRALS 

2.   Each  of  the  integrals  1),  considered  as  functions  of  x^  defined 
over  j^,  have  therefore  upper  and  lower  integrals  in  ^^,  viz.  : 


fdxjfd^        fdxffd% 


For  brevity  these  may  also  be  written 

ff  If  SI  If- 

tLvr-^    ^v.^"^.  ^vy%\  ^vy^c 

733.  1.  Let  f(x-^  ■•■  x„i')  he  limited  in  the  measurable  field  21. 
Let  ^\  be  the  projection  of  2(  on  the  x^-axis.  Let  '^^  be  the  plane 
sections  of  %  corresponding  to  the  points  of  j^.      Then 


ffd^<  fdx^  ffd'^,<  fdx,  ffd^  <  ffdn-,  (1 

ffdn<  fdx  Tfd'^^K  fdx^  f.fdf,<  ffd^.  (2 

Let  us  establish  the  relation  1)  ;  the  demonstration  of  2)  is 
similar. 

Let  21  lie  in  an  outer  cube  ^,  whose  projection  on  the  a;^-axis  is 
b,  and  whose  plane  sections  perpendicular  to  this  axis,  we  denote 
by  O. 

We  introduce  an  auxiliary  function 

g(x^---  x^)  =f(x^  •  •  •  a:^),          at  points  of  21 ; 
=  0,         at  other  points  of  iB. 

Let  I)  he  a.  cubical  division  of  9?^  of  norm  d.  This  divides  ^ 
into  cells  which  we  denote  by  h.  It  also  divides  the  planes  O  into 
cells  which  M^e  denote  by  8'  ;  and  the  segment  b  into  intervals 
which  we  denote  by  h" . 

Let  M,  M'  denote  the  maxima;  and  w,  m'  the  minima  of 
g(x^  •••  x„^  in  the  cells  8,  h'.     Let  G-^  G^  be  the  upper  and  lower 


£1  ^Q  a 

Or,  since 


MULTIPLE   INTEGRALS   TO   ITERATED   INTEGRALS       639 

integrals  of  g  in  the  field  ^.     Let  \f(x-^  ■■■  x„,)\<F  in  21.     Then 
for  each  e  >  0,  there  exists  u  d^  such  that 

a-e<lmh<^Mh<a->t-e,         d<d..  (3 

Moreover,  we  note  that  for  each  x^  of  6, 

m<m'         M' <M, 
SmS'<  fqd^KlMh', 

Multiplying  by  S",  and  summing  over  b,  we  have,  since  8=8'  •  S", 

33        ~  h      ^/q—  SB 
Making  use  of  3),  this  gives 

for  any  cZ<c?o.     Thus,  by  727, 
Or,  since  e  is  small  at  pleasure, 

a<f  (<J  f<a.  (4 

But,  by  725,  _      -ps 

G^=|/cZ2l,         G=\fd%. 

J-  nus  y\  /*  /*      z'  /*      /^ 

j/^2t<j  j  <j  j  <\fd%.  (6 

Let  ^^p  denote  the  upper  content  of  the  frontier  points  of  ^^. 
Then  /»  i 


640  MULTIPLE   PROPER   INTEGRALS 

Let  f 

K^d  ^X/*^^^'         for  points  of  5^ ; 

=  0,         for  other  points  of  b. 
Then,  by  718,  1 ;  730,  and  6), 

But,  by  718,  3, 

)  h(x^dx^  =  )  h(x^dx^  =  j  dx,  ifd'^,. 
In  the  same  way,  we  show 

Thus  1)  has  been  proved., 

2.  In  a  similar  manner  we  may  demonstrate  the  following 
theorem. 

Let  JXx^  •'■  Xfn)  be  limited  in  the  measurable  field  21.  Let  ^^  be  the 
projection  of  '^  on  the  plane  x^  =  0.  Let  a,  be  the  rectilinear  sectioris 
of  21  parallel  to  the  x-axis.      Then 

734.    1.  As  corollaries  of  733  we  have: 

Letf{x-^  '"  Xjn)  be  integrable  in  the  measurable  field  21.      Then 

ifd%  ==SdxSjd^.  =  J^^X-^^^' 

For,  in  this  case,  ^      -^ 

Hence  the  upper  and  lower  integrals  of  j   and  j  are  equal. 


MULTIPLE    INTEGRALS   TO   ITERATED   INTEGRALS       541 

2.   Letf(x-^  ■'•  Xjn)  he  integrahle  in  the  measurable  field  21,  as  well 
as  in  each  of  its  plane  sections  'p.      Then 


X/«=X*J/''*- 


JP/'f  is  integrahle  in  each  of  the  rectilinear  sections  a,  we  also  have 


Sj''^=SM/^-- 


EXAMPLE 
735.   Let  us  find  the  volume  T^of  the  ellipsoid  E, 

^4.^  +  £!  =  l 
a2 "^  62     c2       ' 

By  708,  4,  the  surface  forms  a  discrete  aggregate ;  hence  E  has  a  volume. 
By  702,  2, 


V—\  dzdydz 

JE 

=  B\dx\  dy  dz, 


VFhere  i?'  is  a  quadrant  of  the  ellipse 


F,2      c2  cfi  '~    ~ 


Hence  j\/i_??    ,\/i---^' 

C  /.    »        o2  /•    '        0.1.    611 

\  dydz=\dy        \dz 


=  c  H2/a/ 1  -  ^  -  ^ 


a2      62 


Thus 


F=  2  7r6c  fVl  -^"jfZx  =  f  7ra6c. 


736.  Xe^  /(a^i  •••  2;„,)  ^^  limited  and  integrahle  in  the  measurable 
field  21.  Let  (t>0  he  arbitrarily  small.  Let  j^.  denote  the  points 
ofljorwhich  - 


542  MULTIPLE   PROPER   INTEGRALS 

Then  ^'^  is  discrete.  If  21  is  complete.,  the  upper  content  of  the 
points  where  this  difference  vanishes  is  ^^.  A  similar  theorem  holds 
for  the  differences  -^  ^ 

)  fdx,-  \  fdx^. 

For,  by  734,  1,         ^  r  r      r  r  •     ' 

i   fd^=i   i  =i   i   . 

Hence  f  J -//=/!/-/!  =  0. 

The  theorem  follows  now,  by  731. 

737.  1.  Let  "^  he  a  complete  7neasurable  field.  Let  ^^  he  its  pro- 
jection on  the  x^-axis.  Let  t)^  he  the  points  of  ^^  for  which  the 
corresponding  plane  sections  are  measurahle.      Theyi  X\^  =  ^^. 

A  similar  relatioyi  holds  for  the  projection  2i^. 

'  /(a;j  •••  x„^)=  1,  at  frontier  points  ^  of  21; 

=  0,  at  other  points  of  21. 
Then,  by  702,  -^ 

Since  21  is  measurable,  g"  i^^  discrete,  and  hence  measurable. 
Hence,  by  734,  1,  r      r 

O^jdxl  fd^. 

Hence,  by  731,  the  points  t}^  at  which 


X/^g  =  0 


have  the  same  upper  content  as  ^,. 

2.  Let  f(x-^  •••  x^  he  integrahle  in  the  measurahle  complete  field  21. 
Let  t)^  denote  those  points  of  ^^^  for  ivhich  the  integrals  over  the  corre- 
sponding plane  sections  ^,  exist.  Let  2),  denote  the  points  of  H,  for 
ivh'ch  the  integrals  over  the  corresponding  rectilinear  sections  a^  exist. 
-L  lien  ^  /*        /*  f*         f* 


AFFLiCATlOlN    TO   rNTE-RSIOlM  643 

Let  us  prove  the  first  half  of  1) ;  the  other  half  follows  similarly. 
By  734, 1 

=  Cdx^  f  fd%,        by  726,  737, 


Since 


at  the  points  tf^ 


^?c        ^^. 


Application  to  Inversion 
738.    1.  In  570  we  saw  that 

admits  inversion,  if /(a;,  «/)  is  limited  in  the  rectangle  R=  (ahafi^^ 
and  continuous  except  on  a  finite  number  of  lines  parallel  to  the 
X  and  ?/-axes.      We  can  generalize  this  result  as  follows : 

Let  f(x^  if)  he  limited  in  the  rectangle  R  =  (ahajS^.  Let  the  points 
of  disconti^iuity  A  in  R  he  discrete.  Let  the  points  of  A  on  any  line 
parallel  to  either  axis  form  a  discrete  aggregate  on  that  line.      Then 

j  dy  I  fdx,    I  dx  I  fdy 

exist  and  are  equal. 

For,  by  719,  2,  the  double  integral 

exists.     The  theorem  now  follows  from  734,  2. 

2.  If  the  points  of  discontinuity  oifixy')  on  any  line  parallel  to 
the  x  or  ?/-axes  do  not  form  a  discrete  aggregate,  we  may  apply 
the  following  theorem : 


544  MULTIPLE   PROPER  INTEGRALS 

Let  fixy)  he  limited  in  the  rectangle  R=  (^aba^y 
If  the  points  of  discontinuity  of  f  in  R  form  a  discrete  aggregate^ 
we  have 

\  dy  )  fdx  =  i  dy  I  fdx  =  i  dx  i  fdy  =  \  dx  \  fdy. 

739.  1.  Let  f(x^  y,  z)  he  limited  in  the  rectangular  parallelopiped 
jR,  hounded  hy  the  planes  x=  a,  x=  h;  y  =  a,  y  =  ^\  z  =  A,  z  =  B. 
Let  it  he  continuous  in  -B,  except  at  the  points  of  a  discrete  aggre- 
gate A.  Let  the  jjoints  of  A  on  any  line  or  plane  parallel  to  the  axes 
be  discrete  with  respect  to  that  line  or  plane.  Theti  the  triple  iterated 
integral 

Jdz  \  dy  \  f{xyz')dx 

exists,  and  admits  unrestricted  inversion. 

For, 

}/{xyz-)dB 

exists  by  719,  2.     Let  P  denote  a  plane  section  of  R  parallel  to 
the  a;,  ^/-plane.     Then  the  double  integrals 

{ f(xyz')dP,         A<z<B, 

also  exist  by  719,  2.     Hence  by  734,  2, 

^  fdR  =  Cdz  ffdP. 

^ R  *^ A         ^  P 

But  by  738,      ^  ^^      ^^  ^^      ^^ 

I  fdP  =1   dy)  fdx=  \  dx)  fdy. 

Hence, 


rB  r'p  /»6        ^B  r'b  r'^ 

^  A   ^a    ^a  *^  A  ^a   ^a 


exist,  and  are  equal.     In  the  same  way  we  may  treat  the  four 
other  inversions. 


APPLICATION   TO   INVERSION  545 

EXAMPLES 
740.    1.  Let  i?  =  (0101).     Let 

/(x,  y')  =  0,  for  irrational  x,  or  y; 

=  -  for  X  =  — ;  m,  n  relative  prime,  and  y  rational. 
n  n 

Then 


f  f(xy-)dB 

JR 


exists  by  701,  Ex.  1.    Its  value  is  easily  seen  to  be  0,  by  730. 
We  have  now  : 

J/dx  =  0,    J/rf2/  =  0. 


Also 

Hence 
and  therefore 

On  tbe  other  hand, 
Hence 


I  fdx  =  0,  for  irrational  y,  obviously ; 

=  0  for  rational  y,  by  730. 
\    fdx  =  0,         for  any  y ; 

^\l,£fdx  =  0. 

Cfd,=^.,        iovx=^. 
Jo  n  n 

Cfdy 
Jo 


does  not  exist  for  rational  x;  i.e.  for  a  point  set  which  is  not  discrete. 

However , 

{fdy  =  0 
Jo 

for  any  x.     Hence,  by  734,  1, 

iJ''''=jo'-p'y' 

which  is  obviously  true. 

This  example  illustrates  the  theorem  of  736. 
For,  the  points  x  at  which       _ 

are  the  rational  points  p/q  whose  denominators  q<n. 

Remark.     Let  Bg.  denote  the  content  of  those  points  of  i?  for  which  f(xy)  ^  o- 

Obviously 

Bg-0.     Hence  lim  Ba^  =  0. 

<r=0 

It  would  therefore  be  wrong  to  infer  that 

Cont  21  =  0, 

because  o     *  or        n 

Cont  %^  =  0, 

for  any  <r  >  0. 


546  MULTIPLE   PROPER   INTEGRALS 

2.  Let  E  be  the  rectangle  (0101).  Let  us  suppose  the  coordinates  of  its  pointa 
X,  y  expressed  in  the  dyadic  system.  Cf.  144.  We  define  now  a  jmrtial  aggregate 
%  of  Ji  as  follows.  All  its  points  lie  on  certain  lines  parallel  to  the  (/-axis,  viz.  x  =  a, 
where  0  <  a  <  1  is  any  number  having  a  finite  representation.  Let  a  particular 
value  of  a  embrace  p  digits  in  its  representation.  Those  points  of  the  line  x  =  a 
belong  to  %,  whose  ordinate  is  expressed  inp  digits.  Obviously  this  set  of  points  is 
symmetrical  with  respect  to  x  and  y.  If  the  representation  of  a  is  not  finite,  there 
is  no  point  of  3t  on  the  line  x  =  a,  or  on  y  =  a.  In  any  case,  there  are  but  a  finite 
number  of  points  of  SI  on  any  line  parallel  to  the  x  or  y-axis.  Not  so,  for  lines 
passing  through  a  point  of  %  making  an  angle  of  45°  with  the  a;-axis.  Obviously, 
any  little  segment  of  such  a  line  has  an  infinity  of  points  of  %  in  it.  Thus  %  is 
dense. 

Let  us  define  now  f(xy)  as  follows  : 

f(xy)  =  0,  for  any  point  of  % ; 

=  1,   for  a  point  of  B  not  in  % 
Since  the  oscillation  of  /  in  any  cell  of  i?  is  1,  the  double  integral 

I    fd%       does  not  exist. 
However, 


Hence  both  iterated  integrals 


exist  and  are  equal,     (Pringsheim.) 

3.  In  the  rectangle  B  =  (0101)  let  us  define  another  aggregate  33  as  follows.  As 
before,  the  x  of  every  point  of  i8  must  have  a  finite  representation.  If  the  repre- 
sentation of  a  embraces  p  digits,  all  the  points  of  the  liyie  x  =  a  belong  to  33  whose 
ordinates  are  expressed  by  p  or  less  digits. 

Thus  on  any  given  line  x  =  a,  are  only  a  finite  number  of  points  of  S3.  On  the 
contrary,  on  any  little  segment  of  the  line  y  —  a,  lie  an  infinity  of  points  of  33.  If 
the  representation  of  a  or  &  is  not  finite,  there  is  no  point  of  33  on  the  lines  x=  a, 
OT  y  =  b,  as  in  Ex.  2. 

Let  us  define  f(xy)  as  in  Ex.  2. 

f(xy)  =  0,  for  any  point  in  35 ; 

=  1,  for  a  point  of  i?  not  in  55. 
Then  as  before,  the  double  integral 

does  not  exist. 


k^'"' 


TRANSFORMATION   OF   THE   VARIABLES  647 

The  integral  ^i 

does  not  exist  for  any  y  whose  representation  i^  finite,  since 

On  the  other  hand,  -^ 

J.  fdy  =  \,      for  any  X. 

Hence  /.i         -.^ 

I    dx  I    fdy  —  1.         {Pringsheim.) 

4.  Let/(x2/)  be  limited  in  the  rectangle  B  =  (0101).     Let 

^^'dyf^]f(xy)clx 

exist.     The  reader  might  be  tempted  to  conclude  that  therefore 

C^dyCfdx,        0<«<i3<l,        0<a<b<l, 

Ja         Ja 

exists.    To  show  this  is  not  always  so,  let  us  set  with  Du  Bois  Beymond, 

f(xy)  =  1,  for  rational  y ; 

=  2  x,        for  irrational  y. 
Then  -^ 

I  fdx  =  X,  for  rational  y ; 

Jo' 

=  x^,         for  irrational  y. 
Hence  ^i 

\  fdx  =  \,  for  any  2/; 

and  therefore  -j      ^j 

^/y^/dx  =  i. 

On  the  other  hand,  ^i      ^j 

^^dy^Jdx,         0<6<1, 

does  not  exist,  since  — 

£dy^lfdx  =  h^       j\ly^'fdx  =  b. 

Transformation  of  the   Variables 
741.    1.  Let 

be  defined  over  an  aggregate  %.  These  equations  may  be  re- 
garded as  defining  a  transformation  T,  which  transfers  the  points 
t={t^  •'•  tj^  of  51  to  the  points  u=  (u^  •••  w^).     The  points  t,  u 


548  MULTIPLE   PROPER   INTEGRALS 

may  be  regarded  as  lying  in  the  same  space,  or  in  different  spaces. 
What  we  have  called  the  image  of  2t  defined  by  the  equation  1), 
may  now  be  regarded  as  the  transformed  aggregate  of  31,  and 
may  be  denoted  by  31  y. 

2.  If  the  correspondence  between  the  points  of  31  and  3ty  is 
uniform,  the  equations  1)  admit  as  solutions  the  one-valued  in- 
verse functions 

h  =  ^i(%  •■•  O'  •••  im=  ^mC^i  •••  u^)-  (2 

The  equations  2),  regarded  as  a  transformation,  convert  the 
points  of  Sir  back  to  3t.  For  this  reason  it  is  called  the  inverse 
transformation  of  T,  and  denoted  by  T~^. 

3.  The  transformation 

'^1  =  tj,  •  •  •  Ujji  =  t^ 

is  a  special  case  of  1).  As  it  leaves  every  point  of  31  essentially 
unaltered,  it  is  called  the  identical  transformation,  and  is  denoted 
byl. 

4.  Let 

be  another  transformation  defined  over  31^,  which  we  denote  by  U. 
The  transformation  resulting  from  the  successive  application  of  T 
and  U  is  called  their  product,  and  is  denoted  by  TU.  The  trans- 
formation which  is  applied  first  is  written  first. 

The  result  of  effecting  T,  and  then  its  inverse,  is  to  leave  every 
point  of  31  at  rest.  Hence  TT~^  is  the  identical  transformation; 
in  symbols  ^^_i  ^  ^ 

5.  If  3lr  goes  over  into  31  ^f^  on  applying  U  to  31?',  we  may  regard 
%j.(j  as  the  image  of  31,  afforded  by  the  equations  1),  when  we  con- 
sider the  </)'s  as  functions  of  the  a^'s  through  the  m's,  as  given  by  3) 

6.  The  functions  <^,  -v/r  being  one-valued,  to  any  point  in  31  cor- 
responds one  point  in  31?',  %tu-  Suppose  the  correspondence  between 
3t,  %T(;  is  uniform.  Then  the  correspondence  between  31,  3lr  and 
between  3(2'»  '^tci  is  uniform. 


d<f>. 

d<f>m 

dt. 

dt^ 

501 

TRANSFORMATION   OF   THE   VARIABLES  549 

For,  suppose,  for  example,  that  to  the  point  u  of  Sly  correspond 
two  points  t,  t'  of  St.  If  now  x  corresponds  to  u;  to  the  point  x 
will  correspond  at  least  the  two  points  t,  t'.  The  correspondence 
is  thus  not  uniform  between  SI,  '^ru- 

7.   If  the  functions  ^  have  first  partial  derivatives  in  SI,  we  call 


J    ^5(<^l---  <f>m)  ^ 


the  determinant  of  the  transformation. 

If  the  first  partial  derivatives  of  the  ^'s  are  continuous  in  a 
region  M,  while  the  first  partial  derivatives  of  the  ^|r's  are  con- 
tinuous in  a  region  containing  _By,  we  have,  by  direct  multiplica- 
tion of  the  determinants  t/y,  J^,  and  using  430,  6), 

which  we  may  state  roughly  thus : 

The  determinant  of  the  product  of  two  transformations  is  the  prod- 
uct of  their  determinants. 

742.    1.  Let 

have  continuous  first  partial  derivatives  in  the  region  M.  Let  the 
correspondence  between  M  and  i^y  be  uniform.  Let  the  determi- 
nant of  the  transformation  Jyi^tO  in  M.  In  this  case  we  shall  say 
the  transformation  T  defined  by  the  equations  1)  is  regular  in  R. 

2.  Let  T  he  a  regular  transformation  in  R.  Let  t  he  a  point  of 
R,  to  which  corresponds  the  point  u.  Let  E  he  the  image  of  D^(f). 
There  exists  an  ?;  >  0,  such  that  B  (u)  lies  in  E.  Furthermore^  if 
t  runs  over  an  inner  aggregate  %  of  R,  the  rfs  do  not  sink  helow  some 
positive  number  tjq. 

For,  suppose  there  exists  no  t;  >  0,  such  that  D^(u)  lies  in  E. 
Then  there  exists  a  sequence  of  points  w^,  u^^  •••  which  =  m,  and 


550  MULTIPLE  PEOPER  INTEGRALS 

which  do  not  lie  in  E.  The  inverse  functions  of  1)  being  one- 
valaed  and  continuous  about  m,  by  443,  the  image  of  the  above 
points  form  a  sequence  ^j,  t^^  •••  which  =t.  Hence  all  the  t^  for 
71  >  some  m  lie  in  Dg(t),  and  thus  u^,  ^m+i-,  •••  must  lie  in  S,  which 
is  a  contradiction.  This  establishes  the  first  part  of  the  theorem. 
Turning  to  the  second  part,  suppose  ?;  =  0  as  ^  runs  over  31.  Then 
reasoning  similar  to  that  of  352  leads  at  once  to  a  contradiction. 

3.  Let  T  be  a  regular  transformation  in  the  region  R.  Then  Rj 
is  a  region.  Let  %  he  an  inner  aggregate  of  R.  To  inner  and 
frontier  points  of  St,  correspond  respectively  inner  and  frontier 
points  of  ^  =  Stlrp^  and  conversely. 

This  is  a  direct  consequence  of  2. 

4.  If  T  is  a  regular  transformation  in  the  region  R,  T~^  is  a  regu- 
lar transformation  in  R^.  The  determinant  of  the  inverse  transfor- 
mation is  ^ 

Jrp-\  =  -r- 

This  follows  at  once  from  443  and  741,  4). 

5.  Let  T  he  a  regular  transformation  in  the  region  R.  Let  %  he 
an  inner  aggregate  of  i?,  and  let  ^  he  its  image.  If  either  %  or  ^ 
is  measurable.,  the  other  is.     If  one  is  discrete^  the  other  is. 

This  follows  from  4  and  708,  3. 

743.    1.  Let  T  he  Si  regular  transformation 
T;  x^  =  (f)^(t^---  t,J,  • . .  x,„  =  (f>,^(t^  —  t^')  (1 

in  the  region  R.     Since  Jy^  0,  not  all  the  derivatives 

vanish  at  any  point  of  R.     To  fix  the  ideas,  let 

^^0  (2 

at  a  point  t,  and  hence,  as  it  is  continuous,  in  a  certain  domain 

i>a(0  of  t. 


TRANSFORMATION    OF   THE    VARIABLES  551 

We  show  now  how  T  can  be  expressed  as  the  product  of  two 
special  regular  transformations.  The  first  transformation  we  define 
thus : 

By  virtue  of  2)  this  system  may  be  inverted,  giving 

Here  6  is  one- valued,  and  has  continuous  first  partial  deriva^ 
tives  in  a  certain  domain  I)^(u).  If  h' <h  is  taken  sufficiently 
small,  the  image  U  oi  D^'(t)  lies  in  D^(u). 

We  define  the  transformation  T^  over  U  by 

where  Q  is  the  above  one-valued  function  of  the  w's. 
We  see  at  once  that 

when  t  ranges  over  D^-(f). 

Since  /aj, 

Jrp  =    JtJt^  ^   0, 

it  follows  that  Jj^^^O  in  f/!  Since  the  correspondence  is  uniform, 
and  the  functions  </>,  -yjr  have  continuous  first  derivatives  in  the 
respective  domains,  the  two  transformations  T^,  T^  are  regular. 

2.  Let  %he  a  limited  inner  aggregate  of  the  region  R.  We  can 
effect  a  cubical  division  of  the  t- space  of  norm  d  such  that  for  the 
points  of  51  in  each  cell  d^,  there  exist  two  transformations  2\^*\ 
T^"^  of  the  type  just  considered^  such  that 

For,  we  can  take  d  so  small  that  not  all  the  first  partial  deriva- 
tives vanish  in  any  cell.  For  if  they  did,  reasoning  similar  to 
that  of  264  shows  that  they  must  then  vanish  at  some  point  of  M, 
which  would  require  t7=  0  at  that  point.  Thus  these  cubes  may 
be  taken  as  the  domains  J)sCO  i^^  !•  ^J  reasoning  similar  to  that 
of  352,  we  show  that  the  norms  7}  of  the  domains  D  (u)  considered 


552  MULTIPLE   PROPER   INTEGRALS 

in  1  do  not  sink  below  some  positive  number.  The  same  reasoning 
applied  to  the  norms  B'  of  the  J)s'(t}  above,  shows  that  the  norms  S' 
of  the  above  Z)j.(i),  are  all  greater  than  some  positive  number. 
Thus  if  d  is  taken  small  enough,  the  relation  4)  will  hold  in  each 
cell  containing  points  of  21. 

744.    Let 

define  a  regular  transformation  of  determinant  J  in  the  region  R. 
Let  %  he  any  inner  measurable  perfect  aggregate  of  M,  and  let  3:  be 
its  image.     Letf(x-^  •••  x^')  be  continuous  in  H.      Then 

Jj/(^i  •••  ^m)dx^  "•  dx^=X\J'\fdh  ••'  dt^'  (1 

For  m=\  the  relation  1)  is  easily  seen  to  be  true,  taking 
account  of  direction  in  %.  Let  us  therefore  assume  it  is  correct 
for  w  —  1,  and  show  it  is  so  for  m.  Let  _Z>  be  a  cubical  division  of 
the  t  space,  such  that  in  each  cell  containing  points  of  %  the  trans- 
formation ^can  be  expressed  as  the  product  of  two  transformations 

-^1  '  **!  ==  ^11   "■   '^m-l  =^  '^m-\i  "^m  ==  H>m\P\   '"  tmji 

T^\  X^  —  <^i(Wi  •••  W^_i^),   •••  X^_i  =  </)^_i(Wi  •••  Um-id},  X.„,  =  W^, 

of  the  type  considered  in  743. 

Let  Jj,  e/g  b^  their  determinants.     Then 

If  the  relation  1)  holds  for  each  of  the  partial  aggregates  into 
which  X  falls  after  effecting  2>,  it  obviously  holds  in  3^,  by  728. 
We  may  therefore  assume,  without  loss  of  generality,  that  the 
same  transformations  T^,  T^  may  be  employed  throughout  9£. 

Let  the  image  of  H  in  the  u  space  be  U.     We  have  now 

Lf'^^x  •••  dxra=j  dx„,j  fdx^  •••  dx^_i,  by  737,  2, 

=  I    du„i  I    IJ^lfdu-^^  •••  du„,_i,       by  hypothesis, 
where  ^^,  S^'^n  are  the  transformed  Q,„,  *!)3„„  respectively. 


TRANSFORMATION   OF   THE  VARIABLES  553 

Since  X  is  measurable,  U  is  so,  by  742,  5.     The  same  is  true  of 
^[.     Hence,  by  737,  2, 


j^fdx^  ■  ■ .  dx„,  =  J^  1 J^  I  fdu^  •  • .  du„ 


Applying  the  transformation  T^  to  the  integral  on  the  right, 
similar  considerations  show  that 

j^fdx^  •••  dx„,=jjJ^\\J^\fdtj_  ...  dt^, 
which  is  1). 

745.    Let  ,   ..        .  .  ,    .,       ^  - 

define  a  regular  transformation  of  determmant  J^  in  the  region  R. 
To  a  rectangular  division  D  of  norm  d  of  the  t-space  into  cells  d^, 
corresponds  a  division  A  of  norm  8  of  the  x-space  into  cells  S^.  Let 
%  he  any  inner  region  of  R,  and  X  its  image.  The  cells  of  ^falling 
within  36  are  unmixed,  and  their  contents  are 

^K=\'^\d^  +  ^^d^,         t  in  d^  (1 

where  |  e^  |  <  e  uniformly,  on  taking  d  sufficiently  small. 

For,  %  being  an  inner  region,  to  each  inner  rectangular  cell  d^ 
of  X,  corresponds  a  measurable  cell  S^  of  H  by  742,  5.  Hence  the 
cells  S^  within  H  are  unmixed,  limited,  perfect,  and  finite  in  number 
for  any  A. 

Since  the  determinant  J  is  continuojis  in  St,  we  can  take  d^  so 
small  that  in  any  d^  in  %, 

l^l  =  I^.J  +  ^<;  (2 

where  t^  is  any  point  d^,  and 

for  any  division  I)  of  norm  <  c?q. 

From  702,  2  and  744,  1),  we  have  for  divisions  of  norm  <<?o, 

Cont  h^  =j^dx^...  dx^=:jJJ\dt^  ■■■  dt,„ 

=  K^J  J/^i  •••  dt^+j^<rjt^ ...  dt^,  by  (2, 

where  |eJ<e. 


554  MULTIPLE   PROPER   INTEGRALS 

746.    1.   Let 

T\  X^  =  <^i(^i  •••  t^,   •••  X^  =  ^mih  •••  O 

define  a  regular  transformation  of  determinant  J,  in  a  region  R. 
Let  %  he  any  iymer  aggregate  of  M,  and  let  H  he  its  image.  Let 
/(Xj  •••  Xjn)  he  limited  in  X-      Then 

f^f(X,.:X^-)dl=j^f\j\dZ,  (1 

£f(x,...x„,)di=£f\j\dz.  (2 

For,  let  us  effect  a  cubical  division  of  the  i-space  of  norm  d. 
To  it  corresponds  a  division  of  X  into  cells  of  norm  S. 

Let  us  consider  the  integral  on  the  left  of  1).  Using  the  cus- 
tomary notation,  we  have,  letting  J^  denote  the  value  of  J  at  some 
point  in  <,  ^^^g^  ^  tMXm  +  e,)^.,         by  745,  1), 

=  SM,\J,\d,  +  ^e^M,d,.  (3 

But  if  \f\<Fm  3^, 

\t€,M^d,\<eF(Cm^tZ  +  e)  =  7].  (4 

Let  us  now  consider  the  integral  on  the  right  of  1).     In  the 
^^^^  "^^        Max/-  Min  |J^|<Max  ■/|J^|<Max/-  MaxlJ"], 
if  Max /is  positive  ;  while  the  signs  are  reversed,  if  it  is  negative. 
^^^"^^^^  il!f:  =  Max./|J'|,         ind^. 

^^"^^  M[  =  iHfX  I e/J  +  eO,         I €[  I  <  e  uniformly, 

since  the  oscillation  of  J  in  any  d^  is  uniformly  <  e.     Thus 

^M[d=tM,\J,\d,  +  te[M,d,,  (5 

where,  as  in  4),  \te[MA\<V-  ,  (6 

Thus  3),  5)  give  in  connection  with  4),  6), 
\^M,h^-tM[d^\<2'n, 
which  establishes  the  relation  1).     Similarly  we  may  prove  2). 


TRANSFORMATION   OF   THE   VARIABLES  555 

2.   From  1  a  great  variety  of  theorems  may  be  deduced  by  a 
passage  to  the  limit.     We  note  here  only  the  following : 

Let 
T\  a^i  =  </)i(^i  •  •  •  «^),  •  •  •  2;,„  =  </)„,(^i  . •  •  i^) 

he  continuous  in  the  measurable  aggregate  %^  containing  all  its 
frontier  points  ^ ;  and  regular  in  any  inner  measurable  aggregate  U. 
Let  the  determinant  of  T  he  limited  m  %.  Let  X,  the  image  of  T, 
he  rneasurable  also;  and  let  its  frontier  he  the  image  of  j^.  Let 
f(x^  •■•  x,n)  be  limited  in  X-      Then 


ff(x,.-xjdx=£f\j\dx, 


provided  either  integral  exists. 

For,  let  5)  be  the  image  of  U.     Then,  by  1, 


Let  now  U  =  Xi  then  ?)  =  36. 


INDEX   OF   SOME   TERMS   EMPLOYED 


(Numbers  refer  to  pages.) 


Addition  of  inequalities,  23,  55. 
Adjoint,  integral,  405. 
Archimedean  number  systems,  21,  53. 

Branch,  219. 

principal,  138,  139. 
point,  219. 

Cells,  157,  521. 
Content,  352,  513. 

upper,  lower,  353,  513. 
Continuity/,  208. 

uniform,  215. 

uniform  with  respect  to  a  line,  388 ; 
except  for  certain  points,  390. 

uniform  in  an  interval,  388. 

semi-uniform,  431 ;  in  general,  431. 

regular  in  an  interval,  431 ;  in  gen- 
eral, 431. 
Convergence,  absolute  (of  an  integral), 
405,  445. 

normal  (of  an  integral),  433,  437. 

uniform,  199. 

uniform  with  respect  to  a  line,  388 ; 
except  at  certain  points,  390. 
Correspondence  1  to  1  or  uniform,  133. 

m  to  n,  133. 
Curve,  221. 

arc  of,  221. 

closed,  221. 

multiple  points  of,  221. 

Derivative  of  a  point  aggregate,  162. 
Determinant  of  a  transformation,  549. 


Difference  quotient,  222. 

total,  517. 
Differentials,  first  order,  269. 

higher  order,  277. 
Discontinuity,  211. 

finite,  212. 

infinite,  212. 

removable,  212. 
Distance    between    two    points,    149 ; 
between    two    point    aggregates, 
514. 
Division  of  an  aggregate  or  space,  157, 
521. 

unmixed,  519. 
Domain,  deleted,  154. 

of  definition  of  a  function,  120. 

of  a  point,  153,  195. 

of  a  variable,  119. 

Evanescent,  singular  integral,  401. 

uniformly,  201. 
Extreme,  of  a  variable,  or  rectilinear 
domain,  165,  166. 

isolated,  166. 

point  of  (functions),  317,  322. 

relative  (functions),  329. 

Field  of  integration,  510,  511. 

Forms,  definite,  indefinite,  semidefinite, 

324. 
Frontier,  124. 
Function,  algebraic,  123,  142. 

Beta,  422. 

Cauchy's,  205. 


567 


558 


INDEX 


Function,  composite,  145. 
decreasing,  132. 
Dirichlet's,  204. 

Dirichlet's  definition  of,  120,  143. 
Gamma,  4.53. 
implicit,  282,  283. 
increasing,  132. 

integrable,  336,  400,  404,  445,  511. 
inverse,  133,  135. 
iterated,  160. 

limited,  147  ;  in  general,  399. 
limited  variation,  349,  518. 
monotone,  132. 
primitive,  380. 
totally  differentiable,  269. 
transcendental,  125. 
uni  variant,  132. 

Ideal  points,  172,  194. 

numbers,  173. 
Image,  of  a  number,  21,  79. 

of  a  domain,  146. 
Infinite,  function  is,  213. 
Infinitary,  313. 
Infinity  of  a  function,  213. 
Integra})} e  (integrand  limited),  336,  356, 
511;  absolutely,  405. 

(integrand  infinite),  400,  404 ;  abso- 
lutely, 405. 

(interval  infinite),  445 ;  absolutely, 
445. 
Integra}s,  adjoint,  405. 

convergent,  400,  445 ;  absolutely,  405, 
445. 

definite,  381. 

Euler's,  453. 

Fourier's,  497. 

Fresnel's,  499. 

generalized,  356,  .528. 

improper,  361,  399,  400,  404. 

indefinite,  381. 

iterated,  394,  537,  544. 

lower,  337,  510. 

normally  convergent,  433,  437. 

proper,  361. 


Integrals,    uniformly  convergent,   425, 

465  ;   in  general,  465. 
Stoke's,  463. 
Integration  with  respect  to  a  parameter, 

394. 
Iteration,  160. 

Jacobia7i,  297. 

Limits,  iterated,  198. 

upper  and  lower,  205. 

right  and  left  hand,  172. 

unilateral,  172. 
Limited  functions,  147. 

integrand,  in  general,  399. 

Maxima  and  Minima,  isolated,  166. 

of  a  variable  or  rectilinear  domain, 
165,  166. 

points  of  (functions),  317,  322. 

relative,  329. 
Mean,  law  of,  248. 

first     theorem    of,      366,    417,    459, 
535. 

second  theorem  of,  377,  421,  459. 

value,  167. 
Multipliers,  undetermined,  330. 

Norm,    of    a    division,    157,    336,    50J 
521. 

of  a  domain,  153. 

of  a  vicinity,  155. 
Normal  form  of  a  number,  93. 

singular  integral,  433,  437. 

Order,  of  infinities,  infinitesimals,  313| 
314,  316. 
of  a  point  aggi'egate,  16.3. 
Oscillation  of  a  function,  341,  507. 
Oscillatory  sum,  341,  507. 

Parameter  of  an  integral,  387. 
Parametric  form  of  a  curve,  220. 
Partition,  82. 
Period,  127. 


INDEX 


559 


Points,  at  infinity,  172,  194. 

frontier,  154. 

limiting,  157 ;  proper,  improper,  158  ; 
bilateral,  unilateral,  158. 

ideal,  172,  194. 

isolated,  158. 

outer,  154. 

within  =  inner. 
Point  aggregate,  complement  of,  149. 

complete,  167. 

completed,  .522. 

configurations  of,  149. 

content  of,  3.52,  513 ;  upper,  lower, 
354,  513. 

dense,  167. 

derivative  of,  162. 

difference  of  two,  149. 

discrete,  355,  515. 

distance  between,  514= 

finite,  156. 

frontier  of,  154. 

limited,  156. 

limiting  points  of,  157,  158. 

inner,  515. 

isolated,  167. 

measurable,  353,  513. 

outer,  515. 

partial,  148. 

perfect,  167. 

projection  of,  524,  .525. 

section  of ;   plane,  525 ;    rectilinear, 
525. 

sifted,  .521. 

sLib,  148. 

sum  of,  149. 

union  of,  519. 

unmixed,  519. 
Poles,  123,  142. 
Pnijectloti  of  point  aggregates,  524,  525. 

Region.  167  ;  complete,  167. 
Regular  (integrand  limited),  387. 
(integrand  infinite),  424. 
in    general,   or   except   at   certain 
points,  400. 


Regular   (integrand  infinite),  in   gen- 
eral, or  except  on  certain  lines, 
424,  437. 
simply,    except    on    certain    lines, 
424,  437. 
(interval  infinite),  445,  464. 
in   general,  or   except  on   certain 

points,  445. 
in   general,  or  except  on  certain 

lines,  465. 
simply,    except    on    certain    lines, 
465. 
Rolle's  theorem,  246. 

Scale,  of  infinities,  infiiiitesimals,  312. 

exponential,  315. 

logarithmic,  314. 
Section  plane,  rectilinear,  525. 
Sequence,  24,  61. 

convergent,  25. 

decreasing,  68. 

increasing,  68. 

limited,  68. 

monotone,  68. 

partial,  65. 

regular,  35,  62. 

univariant,  68. 
Singular    Integrals,    relative    to    finite 
points,   401  ;    ideal   point,   447. 

relative    to    finite    lines,    425,    437; 
ideal  Jine,  465. 

evanescent,  401. 

normal,  433,  437. 

uniformly     evanescent,     425,     437, 
465. 
Singular  Lines,  finite,  425,  437;  ideal, 

465. 
Singular    Points,    finite,     399 ;     ideal, 

447. 
Species  of  point  aggregates,  345. 
Successive   Approximation,    method    of 

284. 
System  of  Numbers,  base  of,  92. 

dense,  20,  54. 

dyadic,  triadic,  })i-adic,  92. 


\ 


560 


INDEX 


Total  difference  quotient,  517. 
Totally  differentiable  function,  269. 
Transformation,    of    an    aggregate    or 
space,  .547. 

inverse,  548. 

determinant  of,  549. 


Variation,  of  a  function,  349. 

functions  having  limited,  349,  518. 
Vicinity  of  a  point,  155,  195. 

Within   an   interval,   119 ;    an    m-way 
sphere,  150. 


INDEX   OF   SOME   SYMBOLS   EMPLOYED 

(Numbers  refer  to  pages.) 


(a,h),  (a*,b),  (a,  oo),  etc.,  119. 
D(a),  Dp(a),  Dp*(a),  D(oo),  etc.,  153, 

154,  195. 
F(a),   Vp(a),   Vp*(a),   F(oo),  etc.,  155, 

195. 
Dist  (a,  b),  149 ;  Dist  (5t,  33),  514. 
Max,  165;  Min,  166. 
Mean,  167. 

Cont,  Cont,  Cont,  353,  513. 
lim,  lim  sup,  lim,  lim  inf,  205,  206. 
sgn,  203. 


R,  right  or  right-handed,  155, 172,  206. 
L,  left  or  left-handed,  155,  172,  206. 
U,  unilateral,  172. 
=,  25,  62,  171,  195. 
>,   <,  -,313,314. 

f ,     C,  337,  510. 

Sj),  Sj),  337,  506. 
Md,  ^d,  354,  513. 
21a)  ^A)  521.     (A  general  division.) 


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